---
abstract: 'In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation’s dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $\phi$ possesses certain regularity and sufficient decay as $x \rightarrow \infty$, then the solution $u(t)$ will be smoother than $\phi$ for $0 < t \leq T$ where $T$ is the existence time of the solution.'
author:
- 'Julie Levandosky[^1] Mauricio Sep’[u]{}lveda[^2] Octavio Vera Villagr’[a]{}n[^3]'
title: '**Gain of Regularity for the KP-I Equation**'
---

: KP-I equation, gain in regularity, weighted Sobolev space.

Introduction
============

The KdV equation is a model for water wave propagation in shallow water with weak dispersive and weak nonlinear effects. In 1970, Kadomtsev & Petviashvili [@KP] derived a two-dimensional analog to the KdV equation. Now known as the KP-I and KP-II equations, these equations are given by $$u_{tx} + u_{xxxx} + u_{xx} + \epsilon u_{yy} + (uu_x)_x = 0$$ where $\epsilon = \mp 1$. In addition to being used as a model for the evolution of surface waves [@AC], the KP equation has also been proposed as a model for internal waves in straits or channels of varying depth and width [@Sn], [@DLW]. The KP equation has also been studied as a model for ion-acoustic wave propagation in isotropic media [@PY]. In this paper we consider smoothness properties of solutions to the KP-I equation $$\begin{aligned}
\label{e101}&  & (u_{t} + u_{xxx} + u_{x} + u\,u_{x})_{x} -
u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\
\label{e102}&  & u(x,\,y,\,0)=\phi(x,\,y).\end{aligned}$$

Certain results concerning the Cauchy problem for the KP-I equation include the following. Ukai [@Uk] proved local well-posedness for both the KP-I and KP-II equations for initial data in $H^s(\mathbb R^2)$, $s \geq 3$, while Saut [@Sa] proved some local existence results for generalized KP equations. More recently, results concerning global well-posedness for the KP-I equation have appeared. In particular, see the works of Kenig [@Ke] and Molinet, Saut, and Tzvetkov [@MST]. Here we consider the question of gain of regularity for solutions to the KP-I equation.

A number of results concerning gain of regularity for various nonlinear evolution equations have appeared. This paper uses the ideas of Cohen [@Co], Kato [@Ka], Craig and Goodman [@CG] and Craig, Kappeler, and Strauss [@CKS]. Cohen considered the KdV equation, showing that “box-shaped" initial data $\phi \in L^2(\mathbb R^2)$ with compact support lead to a solution $u(t)$ which is smooth for $t > 0$. Kato generalized this result, showing that if the initial data $\phi$ are in $L^2((1+e^{\sigma x})\,dx)$, the unique solution $u(t) \in C^\infty(\mathbb R^2)$ for $t > 0$. Kruzhkov and Faminskii [@KF] replaced the exponential weight function with a polynomial weight function, quantifying the gain in regularity of the solution in terms of the decay at infinity of the initial data. Craig, Kappeler, and Strauss expanded on the ideas from these earlier papers in their treatment of highly generlized KdV equations.

Other results on gain of regularity for linear and nonlinear dispersive equations include the works of Hayashi, Nakamitsu, and Tsutsumi [@HNT1], [@HNT2], Hayashi and Ozawa [@HO], Constantin and Saut [@CS], Ponce [@Po], Ginibre and Velo [@GV], Kenig, Ponce and Vega [@KPV], Vera [@thesi1], [@Ve] and Ceballos, Sepulveda and Vera [@CSV].

In studying propagation of singularities, it is natural to consider the bicharacteristics associated with the differential operator. For the KdV equation, it is known that the bicharacteristics all point to the left for $t > 0$, and all singularities travel in that direction. Kato [@Ka] makes use of this uniform dispersion, choosing a nonsymmetric weight function decaying as $x \rightarrow -\infty$ and growing as $x \rightarrow \infty$. In [@CKS], Craig, Kappeler and Strauss also make use of a unidirectional propagation of singularities in their results on infinite smoothing properties for generalized KdV-type equations for which $f_{u_{xxx}} \geq c > 0$.

For the two-dimensional case, Levandosky [@Le1] proves smoothing properties for the KP-II equation. This result makes use of the fact that the bicharacteristics all point into one half-plane. Subsequently, in [@Le2], Levandosky considers generalized KdV-type equations in two-dimensions, proving that if all bicharacteristics point into one half-plane, an infinite gain in regularity will occur, assuming sufficient decay at infinity of the initial data.

In this paper, we address the question regarding gain in regularity for the KP-I equation. Unlike the KP-II equation, the bicharacteristics for the KP-I equation are not restricted to a half-plane but span all of $\mathbb R^2$. As a result, singularities may travel in all of $\mathbb R^2$. However, here we prove that if the initial data decays sufficiently as $x \rightarrow \infty$, then we will gain a finite number of derivatives in $x$ (as well as mixed derivatives). In order to state a special case of our gain in regularity theorem, we first introduce certain function spaces we will be using.\
We define $$\begin{aligned}
\label{e106}X^{0}(\mathbb{R}^{2})=
\left\{u:\;u,\;\xi^{3}\widehat{u},\;\frac{\eta^{2}}{\xi}\,\widehat{u}\in
L^{2}(\mathbb{R}^{2})\right\}\end{aligned}$$ equipped with the natural norm. On the space $$\begin{aligned}
\label{e107}\widetilde{X}^{0}(\mathbb{R}^{2})=
\left\{u:\;\frac{1}{\xi}\,\widehat{u}(\xi,\,\eta)\in
L^{2}(\mathbb{R}^{2})\right\}\end{aligned}$$ we define the operator $\partial_{x}^{-1}$ by $\widehat{\partial_{x}^{-1}u}\equiv\frac{1}{i\,\xi}\,\widehat{u}.$ Therefore, in particular, we can write the norm of $X^{0}(\mathbb{R}^{2})$ as $$\begin{aligned}
\label{e108}||u||_{X^{0}(\mathbb{R}^{2})}^{2}=\int_{\mathbb{R}^{2}}[\,u^{2}
+ u_{xxx}^{2} + (\partial_{x}^{-1}u_{yy})^{2}\,]\,dx\,dy<+\infty\end{aligned}$$ On this space of functions $X^{0}(\mathbb{R}^{2}),$ it makes sense to rewrite - as $$\begin{aligned}
\label{e109}&  & u_{t} + u_{xxx} + u_{x} + u\,u_{x} -
\partial_{x}^{-1}u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\
\label{e110}&  & u(x,\,y,\,0)=\phi(x,\,y)\end{aligned}$$ and consider weak solutions $u\in X^{0}(\mathbb{R}^{2}).$\
\
[*Definition.*]{} Let N be a positive integer. We define the space of functions $X^{N}(\mathbb{R}^{2})$ as follows $$\begin{aligned}
\label{e111}X^{N}=\left\{u:\;u\in L^{2}(\mathbb{R}^{2}),\;{\cal
F}^{-1}(\xi^{3}\,\widehat{u})\in H^{N}(\mathbb{R}^{2}),\,{\cal
F}^{-1}\left(\frac{\eta^{2}}{\xi}\,\widehat{u}\right)\in
H^{N}(\mathbb{R}^{2})\right\}\end{aligned}$$ equipped with the norm $$\begin{aligned}
\label{e112}||u||_{X^{N}(\mathbb{R}^{2})}^{2} =
\int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|\alpha|\leq
N}[\,(\pa u_{xxx})^{2} +
(\pa \partial_{x}^{-1}u_{yy})^{2}\,]\right)\,dx\,dy<+\,\infty\end{aligned}$$ where $\alpha=(\alpha_1,\,\alpha_2)\in\mathbb{Z}^{+}\times\mathbb{Z}^{+}$ and $|\alpha|=\alpha_1 + \alpha_2.$\
\
\

(1) If we consider $|\alpha| = 2L-1$ above, then $\alpha = (2L-1,0)$ in which case the result states that $$\sup_{0 \leq t \leq T} \int_{\mathbb R^2} t^{L-1} (x_+ + e^{\sigma x_-})(\partial_x^{2L-1}u)^2
    + \int_0^T \int_{\mathbb R^2} t^{L-1} (1 + e^{\sigma x_-})(\partial_x^{2L}u)^2 < \infty.$$ In particular, this result shows a [*gain*]{} in $L$ derivatives in $x$.

(2) While we gain $x$ derivatives and mixed derivatives, we do not gain pure $y$ derivatives. However, we do not require any weighted estimates on $\partial_y^L u$. In addition, we do not require any weighted estimates on $u$. The results on the KP-II equation include gains in pure $y$ derivatives, but also require weighted estimates on $\partial_y^L u$.

(3) The assumptions on $u$ are reasonable and shown to hold in section 6.

The main idea of the proof is the following. We use an inductive argument where on each level $|\alpha|$, we apply the operator $\pa = \partial_x^{\alpha_1}
\partial_y^{\alpha_2}$ to , multiply the differentiated equation by $2f_\alpha\pa u$ where $f_\alpha$ is our weight function, to be specified later, and integrate over $\mathbb R^2$. Doing so, we arrive at the following inequality $$\begin{split}
\label{main-eq}
& \partial_t \int f (\pa u)^2 + 3 \int f_x (\pa u_x)^2 \leq \int f_x (\pa \partial_x^{-1} u_y)^2 \\
& \qquad + \int [f_t + f_{xxx} + f_x] (\pa u)^2 + \left|2 \int f (\pa u)\pa (uu_x)\right|
\end{split}$$ where $\int = \int_{\mathbb R^2} dx dy$. Assuming $f_x > 0$, the second term on the left-hand side has a positive sign, thus allowing us to prove a gain in regularity. We notice that the first term on the right-hand side is of order $|\alpha|$. By choosing appropriate weight functions for each $\alpha$, we have a bound on that term from the previous step of the induction. After proving estimates involving the nonlinear term on the right-hand side of the equation, we apply Gronwall’s inequality to prove the bounds on the terms on the left-hand side of the equation.

The plan of the paper is the following. In section 2 we show the derivation of . In sections 3 and 4 we prove an existence result showing that for initial data $\phi \in X^N(\mathbb R^2)$ there exists a smooth solution $u \in L^\infty([0,T];
X^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{X^0}$. In section 5 we prove estimates for the terms on the right-hand side of . In section 6 we prove a priori estimates showing that the solution $u$ found in section 4 also satisfies for the same time $T$ as long as $$\int (\phi^2 + (\partial_y^L \phi)^2 + (1+x_+)^L(\partial_x^L \phi)^2) < \infty.$$ Once we have found the solution $u$ in the appropriate weighted space as well as bounds for terms on the right-hand side of , in section 7, we can state and prove our main gain in regularity result. This proof uses an inductive argument along with the main estimates proven in section 5.\
We will be using non-symmetric weight functions. In particular, we will be using weight functions $f(x,t) \in C^\infty$ which behave roughly like powers of $x$ for $x > 1$ and decay exponentially for $x < -1$. We define our weight classes as follows.\
A function $f = f(x,\,t)$ belongs to the weight class $W_{\sigma \;i\;k}$ if it is a positive $C^{\infty }$ function on $\mathbb{R}\times [0,\,T]$ and there are constants $c_{j},\,0\leq j\leq 5$ such that $$\begin{aligned}
\label{e103}&0<c_{1}\leq t^{-\,k}\,e^{-\,\sigma \,x}\,f(x,\,t)
\leq c_{2}\qquad \forall
\;x<-1,\quad 0<t<T.&\\
\label{e104}&0<c_{3}\leq t^{-\,k}\,x^{-\,i}\,f(x,\,t)\leq
c_{4}\qquad \forall
\;x>1,\quad 0<t<T.&\\
\label{e105}&\left(t\mid \partial_{t}f\mid + \mid
\partial_{x}^{r}f\mid \right)/f\leq c_{5}\qquad \forall
\;(x,\,t)\in \mathbb{R}\times [0,\,T],\quad \forall \;r\in
\mathbb{N}.&\end{aligned}$$ Thus $f$ looks like $t^{k}$ as $t\rightarrow 0,$ like $x^{i}$ as $x\rightarrow +\infty$ and like $e^{\sigma\,x}$ as $x\rightarrow
-\infty.$\
\
Before proceeding, we introduce some other function spaces we will be using.\
Let $N$ be a positive integer. Let $\widetilde H^{N}_x(W_{\sigma\;i\;k})$ be the space of functions $$\begin{aligned}
\label{e113}\widetilde H^{N}_x(W_{\sigma \;i\;k})= \left\{v\colon
\mathbb{R}^{2}\rightarrow \mathbb{R}\,:\,\;\,||v||_{\widetilde
H^{N}_x(W_{\sigma \;i\;k})}^{2}= \int \sum_{|\alpha| \leq N} [(\pa
v)^2 + f|\partial_x^{N} v|^{2}] <+\,\infty \,\right\}\end{aligned}$$ with $f\in W_{\sigma \;i\;k}$ fixed.\
\
[**Remarks.**]{}

(1) We note that although the norm above depends on $f,$ all choices of $f$ in this class lead to equivalent norms.

(2) The usual Sobolev space is $\,H^{N}(\mathbb{R}^{2})$ without a weight.

For fixed $f\in W_{\sigma \;i\;k}$ define the space ($N$ be a positive integer) $$\begin{aligned}
\lefteqn{L^{2}([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}\nonumber \\
\label{e114}& = &
  \left\{v(x,\,y,\,t):\;
  ||v||_{L^{2}([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}^{2}= \int_{0}^{T}||v(\,\cdot
\,,\,\cdot\,,\,t)||_{\widetilde H_x^{N}(W_{\sigma \;i\;k})}^{2}dt<+\,\infty
\,\right\}\end{aligned}$$ $$\begin{aligned}
\lefteqn{L^{\infty }([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k})) }
\nonumber \\
\label{e115}& = & \left\{\,v(x,\,y,\,t):\;||v||_{L^{\infty
}([0,\,T]:\,\widetilde H_x^{N}(W_{\sigma \;i\;k}))}= \sup_{t\in
[0,\,T]}\,||v(\,\cdot\,,\,\cdot \,,\,t)||_{\widetilde H_x^{N}(W_{\sigma
\;i\;k})}<+\,\infty \,\right\}.\end{aligned}$$ For simplicity, let $$\begin{aligned}
\label{e118}{\cal Z}_{L} = X^{1}(\mathbb{R}^{2})\bigcap
\widetilde{H}_x^{L}(W_{0\;L\;0}).\end{aligned}$$ With this notation, ${\cal Z}_{L}$ consists of those functions $u$ such that $$\begin{aligned}
\label{e119}||u||_{{\cal
Z}_{L}}^{2}=\int_{\mathbb{R}^{2}}[u^2 +
u_{xxxx}^{2} + (\partial_{x}^{-1}u_{yy})^{2} + u_{yy}^2 + \sum_{|\alpha| \leq L}
(\pa u)^2 + f(\partial_x^L u)^2 \,]\,dx\,dy\end{aligned}$$ for some $f\in W_{0\;L\;0}.$\
\
We now state a lemma describing one of the types of bounds we will be using for our a priori estimates.\
$$\begin{aligned}
\label{e120}||u||_{L^{\infty}(\mathbb{R}^{2})}\leq
c\left(\int_{\mathbb{R}^{2}}[\,1 + |\xi|^{p} +
|\eta|^{q}\,]\,|\widehat{u}|^{2}\,d\xi\,d\eta\right)^{1/2}.\end{aligned}$$ [*Proof.*]{} The proof follows from writing $u$ in terms of its inverse Fourier transform and using the fact that $$\begin{aligned}
\int_{\mathbb{R}^{2}}\frac{1}{1 + |\xi|^{p} +
|\eta|^{q}}\,d\xi\,d\eta <+\infty\end{aligned}$$ for $p,\,q$ satisfying our hypothesis. $\square$\
\
In particular, we have: $$\begin{aligned}
\label{e121}||u||_{L^{\infty}(\mathbb{R}^{2})}\leq
c\left(\int_{\mathbb{R}^{2}}[\,u^{2} + u_{xx}^{2} +
u_{y}^{2}\,]\,dx\,dy\right)^{1/2}.\end{aligned}$$

Main Equality
=============

We consider the KP-I equation $$\begin{aligned}
\label{e201}&  & u_{t} + u_{xxx} + u_{x} + u\,u_{x} -
\partial_{x}^{-1}u_{yy} =0,\qquad (x,\,y)\in\mathbb{R}^{2},\quad t\in\mathbb{R}\\
\label{e202}&  & u(x,\,y,\,0)=\phi(x,\,y).\end{aligned}$$ [**Lemma 2.1.**]{} [*Let $u$ be a solution of - with enough Sobolev regularity and with sufficient decay at infinity. Let $f=f(x,t)$. Then*]{} $$\begin{aligned}
\lefteqn{\partial_{t}\int_{\mathbb{R}^{2}}f
(\pa u)^{2}\,dx\,dy +
\int_{\mathbb{R}^{2}}g\,(\pa u_x)^{2}\,dx\,dy}\nonumber \\
\label{e203}&  & +\int_{\mathbb{R}^{2}}\theta\,
(\pa u)^{2}\,dx\,dy +
\int_{\mathbb{R}^{2}}\theta_{1}\, (\pa \partial_x^{-1} u_y)^{2}\,dx\,dy +
\int_{\mathbb{R}^{2}}R_{\alpha}\,dx\,dy = 0\end{aligned}$$ [*such that*]{} $$\begin{aligned}
g & = & 3\,f_{x}\\
\theta & = & -\;[f_{t} + f_{xxx} + f_{x}\,]\\
\theta_{1} & = & -\;f_{x}\\
R_{\alpha} & = & 2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2}
{\alpha_1\choose n}{\alpha_2\choose m}f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u)
\,(\partial_{x}^{\alpha_1 + 1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$ [*Proof.*]{} Applying the operator $\pa$ to , we have $$\begin{aligned}
&  & \pa u_{t} + \pa u_{xxx} + \pa u_x + \pa (u\,u_x) -
\pa \partial_x^{-1} u_{yy} =0.\end{aligned}$$ Multiplying by $2\,f\,
\pa u$ and integrating over $\mathbb{R}^{2},$ we have $$\begin{split}
\label{e204}
& 2\int f\,(\pa u)\,(\pa u)_{t} +
2\int f\,(\pa u)\,(\pa u_{xxx})
+ 2\int f\,(\pa u)\,(\pa u_x) \\
& \qquad + 2\int f\,
(\pa u)\,\pa (u\,u_{x}) -\;2\int f\,
(\pa u)\,(\pa \partial_{x}^{-1} u_{yy}) =0.
\end{split}$$ Each term in is calculated separately integrating by parts $$\begin{aligned}
2\int f\,
(\pa u)\,(\pa u)_{t} = \partial_{t}\int f\,(\pa u)^{2} -
\int f_{t}\,(\pa u)^{2}.\end{aligned}$$ $$\begin{aligned}
2\int f\,(\pa u)\,(\pa u_{xxx})
=3\int f_{x}\,(\pa u_x)^{2} -
\int f_{xxx}\,(\pa u)^{2}.\end{aligned}$$ $$\begin{aligned}
2\int f\,(\pa u)\,(\pa u_x) =
-\int f_{x}\,(\pa u)^{2}\end{aligned}$$ $$\begin{aligned}
-\;2\int f\,
(\pa u)\,(\pa \partial_{x}^{-1}u_{yy}) =-\int f_{x}\,
(\pa \partial_{x}^{- 1}u_y)^{2}.\end{aligned}$$ $$\begin{aligned}
2\int f\,
(\pa u)\,\pa (u\,u_{x}) = \;2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2}{\alpha_1\choose
n}{\alpha_2\choose
m}\int f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 +
1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$ Replacing in we obtain $$\begin{split}
& \partial_{t}\int f\,(\pa u)^{2} +
3\int f_{x}\,(\pa u_x)^{2} \\
& - \int [f_{t} + f_{xxx} + f_{x}]\,(\pa u)^{2} -
\int f_{x}\,(\pa \partial_{x}^{-
1}u_y)^{2} \\
& +\;2\sum_{n=0}^{\alpha_1}\sum_{m=0}^{\alpha_2}{\alpha_1\choose
n}{\alpha_2\choose m}\int f\,(\pa u)\,(\partial_{x}^{n}
\partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 + 1
- n}\partial_{y}^{\alpha_2 - m}u) = 0.
\end{split}$$ Therefore, we obtain the [**Main Equality**]{}, $$\begin{split}
& \partial_{t}\int f
(\pa u)^{2} +
3\int f_{x}\,(\pa u_x)^{2} + \int \theta\,
(\pa u)^{2} +
\int \theta_{1}\, (\pa \partial_{x}^{-
1}u_y)^{2} +
\int R_{\alpha} = 0
\end{split}$$ such that $$\begin{aligned}
\theta & = & -\;[f_{t} + f_{xxx} + f_{x}\,]\\
\theta_{1} & = & -\;f_{x}\\
R_{\alpha} & = & 2\sum_{n=1}^{\alpha_1}\sum_{m=1}^{\alpha_2}{\alpha_1\choose n}{\alpha_2
\choose m}f\,(\pa u)\,(\partial_{x}^{n}\partial_{y}^{m}u)\,(\partial_{x}^{\alpha_1 +
1 - n}\partial_{y}^{\alpha_2 - m}u).\end{aligned}$$

An a priori estimate
====================

In section four we prove a basic local-in-time existence theorem for -. The proof relies on approximating by a sequence of linear equations. In this section, we prove an existence theorem for linear equations as well as an a priori estimate on those solutions which will be necessary for our main existence theorem in the next section.

We begin by approximating by the linear equation $$\begin{aligned}
\label{e305}&  & u_{t}^{(n)} + u_{xxx}^{(n)} + u_{x}^{(n)} + u^{(n
- 1)}\,u_{x}^{(n)} -
\partial_{x}^{-1}u_{yy}^{(n)} =0\end{aligned}$$ where the initial condition is given by $u^{(n)}(x,\,y,\,0)=\phi(x,\,y)$ and the first approximation is given by $u^{(0)}(x,\,y,\,t)=\phi(x,\,y).$ The linear equation which is to be solved at each iteration is of the form $$\begin{aligned}
\label{e306}&  & u_{t} + u_{xxx} + u_{x} + b\,u_{x} -
\partial_{x}^{-1}u_{yy} =0.\end{aligned}$$ where $b$ is a smooth bounded coefficient. Below we show that this equation can be solved in any interval of time in which the coefficient is defined.\
(Existence for linear equation). [*Given initial data $\phi\in H^{\infty}(\mathbb{R}^{2})=\bigcap_{N\geq
0}H^{N}(\mathbb{R}^{2})$ and $\partial_{x}^{-1}\phi_{yy}\in\bigcap_{N\geq
0}H^{N}(\mathbb{R}^{2})$ there exists a unique solution of . The solution is defined in any time interval in which the coefficients are defined.*]{}\
\
[*Proof.*]{} Let $T>0$ be arbitrary and $M>0$ be a constant. Let $$\begin{aligned}
{\cal L} = \partial_{t} + \partial_{x}^{3} + \partial_{x} +
b\,\partial_{x} - \partial_{x}^{-1}\partial_{y}^{2}\end{aligned}$$ be defined on those functions $u\in X^{0}(\mathbb{R}^{2}).$ Recall that $u\in X^{0}(\mathbb{R}^{2})$ means $u,\,u_{xxx},\,\frac{\eta^{2}}{\xi}\widehat{u}\in
L^{2}(\mathbb{R}^{2}).$ We consider the bilinear form ${\cal
B}:{\cal D}\times {\cal D}\rightarrow \mathbb{R},$ $$\begin{aligned}
{\cal B}(u,\,v) =
\left<u,\,v\right>=\int_{0}^{T}\int_{\mathbb{R}^{2}}e^{-M\,t}\,u\,v\,dx\,dy\,dt\end{aligned}$$ where ${\cal D}=\{u\in
C([0,\,T]:\,L^{2}(\mathbb{R}^{2})):\,u(x,\,y,\,0)=0\}.$ By integration by parts, we see that $$\begin{aligned}
\int_{\mathbb{R}^{2}}{\cal L}u\cdot u\,dx\,dy & = &
\frac{1}{2}\,\partial_{t}\int_{\mathbb{R}^{2}}u^{2}\,dx\,dy -
\frac{1}{2}\int_{\mathbb{R}^{2}}b_{x}\,u^{2}\,dx\,dy \\
& \geq & \frac 12 \partial_t \int_{\mathbb R^2} u^2 \,dx\,dy - \frac 12 \int_{\mathbb R^2}
c u^2 \,dx\,dy\end{aligned}$$ We multiply by $e^{-M\,t}$ and integrate in time to obtain for $u\in C([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ with $u(x,\,y,\,0)=0$ $$\begin{aligned}
\label{e307}\left<{\cal L}u,\,u\right> \geq
e^{-M\,t}\int_{\mathbb{R}^{2}}u^{2}\,dx\,dy + (M -
c)\int_{\mathbb{R}^{2}}e^{-M\,t}\,u^{2}\,dx\,dy.\end{aligned}$$ Thus, $\left<{\cal L}u,\,u\right>\geq \left<u,\,u\right>$ provided $M$ is chosen large enough. Similarly, $\left<{\cal L}^{*}v,\,v\right>\geq \left<v,\,v\right>$ for all $v\in
C([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ such that $v(x,\,y,\,T)=0$ where ${\cal L}^{*}$ denotes the formal adjoint of ${\cal L}.$ Therefore, $\left<{\cal L}^{*}v,\,{\cal L}^{*}u\right>$ is an inner product on ${\cal
D}^{*}=\{v\in
C([0,\,T]:\,X^{0}(\mathbb{R}^{2})):\,v(x,\,y,\,T)=0\}.$ Denote by $Y$ the completion of ${\cal D}^{*}$ with respect to this inner product. By the Riesz representation theorem, there exists a unique solution $V\in Y,$ such that for any $v\in {\cal D}^{*},$ $\left<{\cal
L}^{*}V,\,{\cal L}^{*}v\right>=(\phi,\,v(x,\,y,\,0))$ where we used the fact that $(\phi,\,v(x,\,y,\,0))$ is a bounded linear functional on ${\cal D}^{*}.$ Then $w={\cal L}^{*}V$ is a weak solution of ${\cal
L}w=0,$ $w_{0}=\phi$ with $w\in
L^{2}(\mathbb{R}^{2}\times [0,\,T]).$

[*Remark.*]{} To obtain higher regularity of the solution, we repeat the proof with higher derivatives included in the inner product. It is a standard approximation procedure to obtain a result for general initial data. $\square$\
\
Next, we need to introduce a new function space. Let $$\begin{aligned}
\label{e303}{Z}_{T}^{N}=\{u:\;u\in L^{\infty}([0,\,T]:\,H^{(N +
3,\,N + 2)}(\mathbb{R}^{2})),\;u_{t}\in
L^{\infty}([0,\,T]:\,H^{N}(\mathbb{R}^{2}))\}\end{aligned}$$ where $H^{(\alpha_1,\,\alpha_2)}(\mathbb{R}^{2})=\{u:\;u,\,
\partial_{x}^{\alpha_1}u,\,\partial_{y}^{\alpha_2}u\in
L^{2}(\mathbb{R}^{2})\}$ with the accompanying norm $$\begin{aligned}
\label{e304}||u||_{{Z}_{T}^{N}}^{2}=\sup_{t\in
[0,\,T]}\int_{\mathbb{R}^{2}}\left(u^{2} +
\sum_{|j|=N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}u_{yy})^{2}\,]\right)+
\int_{\mathbb{R}^{2}}\left(u_{t}^{2} +
\sum_{|j|=N}(\partial^{j}u_{t})^{2}\right).\end{aligned}$$

Using this function space and the linearized equation , we consider the mapping $\Pi: Z_{T}^{N} \rightarrow Z_{T}^{N}$ such that $u^{(n)} = \Pi(u^{(n-1)})$ and our first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. In Lemma 3.2 below, we show an a priori estimate which will be used on our sequence of solutions $\{u^{(n)}\}$ in our main existence theorem in section four.\
\
[**Lemma 3.2.**]{} [*Let $v,\,w$ be a pair of functions in ${Z}_{t}^{N}$ for all $N$ and all $t\geq 0,$ such that $v,\,w$ are solutions to*]{} $$\begin{aligned}
\label{e308}&  & v_{t} + v_{xxx} + v_{x} + w\,v_{x} -
\partial_{x}^{-1}v_{yy} =0.\end{aligned}$$ [*Then for all $N\geq 0,$ the following inequality holds: $$\begin{aligned}
\label{e309}||v||_{{Z}_{t}^{N}}^{2} \leq
||v(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N +
2)}(\mathbb{R}^{2})}^{2} +
||v_{t}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2} +
c\,t\,||w||_{{Z}_{t}^{N}}\;||v||_{{Z}_{t}^{N}}^{2}\end{aligned}$$ for all $t\geq 0.$*]{}\
\
[*Proof.*]{} We will show that for each $j,$ $|j|\geq 0$ and $0\leq \widetilde{t}\leq t,$ $$\begin{aligned}
\partial_{t}\int[\,(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}
+
& (\partial^{j}\partial_{x}^{3}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}
+
(\partial^{j}\partial_{y}^{2}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}
+
(\partial^{j}v_{t}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}\,] \\
& \leq
c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ We begin by taking $j$ derivatives of . We have $$\begin{aligned}
\label{e310}&  & \partial^{j}v_{t} + \partial^{j}v_{xxx} +
\partial^{j}v_{x} + \partial^{j}(w\,v_{x}) -
\partial^{j}\partial_{x}^{-1}v_{yy} =0.\end{aligned}$$ Multiply by $2\,\partial^{j}v$ and integrate over $\mathbb{R}^{2}.$ Hence $$\begin{aligned}
\partial_{t}\int
(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}  &
\leq &
c\,\left|\int\partial^{j}(w\,v_{x})\,(\partial^{j}v) \right|\\
& \leq &\left|\int[\,(\partial^{j}w)\,v_{x}\;+\,
\ldots\,+\;w\,(\partial^{j}v_{x})\,]\,(\partial^{j}v) \right|.\end{aligned}$$ The remainder terms can be bounded as follows: $$\begin{aligned}
\left|\int(\partial^{j}w)\,v_{x}\,(\partial^{j}v) \right|
& \leq & ||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}w)^{2} \right)^{1/2}
\left(\int(\partial^{j}v)^{2} \right)^{1/2}\\
& \leq & c\left(\int[\,v_{x}^{2} + v_{xxx}^{2} +
v_{xy}^{2}\,] \right)^{1/2}||w||_{H^{|j|}(\mathbb{R}^{2})}\;
||v||_{H^{|j|}(\mathbb{R}^{2})}\\
& \leq & c\;||w||_{{Z}_{t}^{|j|}}\; ||v||_{{Z}_{t}^{|j|}}^{2}\end{aligned}$$ and $$\begin{aligned}
\left|\int w\,(\partial^{j}v_{x})\,(\partial^{j}v) \right|
& \leq &
\left|\int w_{x}\,(\partial^{j}v)^{2} \right|
\\
& \leq &
c\;||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int
(\partial^{j}v)^{2} \\
& \leq & c\left(\int[\,w_{x}^{2} + w_{xxx}^{2} +
w_{xy}^{2}\,] \right)^{1/2}||v||_{{Z}_{t}^{|j|}}^{2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Therefore, we obtain $$\begin{aligned}
\partial_{t}\int(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} 
\leq c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next, we take three $x$ derivatives of , multiply by $2\,\partial^{j}v_{xxx}$ and integrate over $\mathbb{R}^{2}.$ Our inequality becomes $$\begin{aligned}
\partial_{t}\int\partial^{j}v_{xxx}
& \leq  c\left|\int(\partial^{j}(w\,v_{x})_{xxx})
\,(\partial^{j}v_{xxx})  \right|\\
& \leq  c\left|\int
\partial^{j}(w_{xxx}\,v_{x} + 2\,w_{xx}\,v_{xx} + 2\,w_{x}\,v_{xxx} + w\,v_{xxxx})
(\partial^{j}v_{xxx})  \right|\\
& \leq  c\left|\int
\partial^{j}(w_{xxx}\,v_{x})\,(\partial^{j}v_{xxx}) \right|
+
c\left|\int\partial^{j}(w_{xx}\,v_{xx})\,
(\partial^{j}v_{xxx}) \right|\\
&  \qquad +\; c\left|\int\partial^{j}(w_{x}\,v_{xxx})\,
(\partial^{j}v_{xxx}) \right| + c
\left|\int\partial^{j}(w\,v_{xxxx})\,(\partial^{j}v_{xxx}) 
\right|\\
& \leq I_{1} + I_{2} + I_{3} + I_{4}.\end{aligned}$$ We will look at terms $I_{k},\;k=1,\,2,\,3,\,4$ below. For $I_{1}$ we have $$\begin{aligned}
\lefteqn{\left|\int
\partial^{j}(w_{xxx}\,v_{x})\,(\partial^{j}v_{xxx}) \right|}\\
& = & \left|\int
[\,\partial^{j}w_{xxx})\,v_{x}\,(\partial^{j}v_{xxx})\;+ \,\ldots
\,+\;
w_{xxx}\,(\partial^{j}v_{x})\,(\partial^{j}v_{xxx})\,] \right|\\
& \leq & ||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}w_{xxx})^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
&  & +\, \cdots\, +
||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int w_{xxx}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
& \leq & ||v||_{{Z}_{t}^{0}}\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}} +\,\cdots \, +
||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}\\
& \leq & c\;||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{2},$ $$\begin{aligned}
\left|\int\partial^{j}(w_{xx}\,v_{xx})
\,(\partial^{j}v_{xxx}) \right| =
\left|\int[\,(\partial^{j}w_{xx})\,v_{xx}
+\,\ldots\,+
w_{xx}(\partial^{j}v_{xx})\,]\,(\partial^{j}v_{xxx}) \right|.\end{aligned}$$ To bound these terms, we will use the following anisotropic imbedding in [@BIN]. For $2\leq n<6,$ $$\begin{aligned}
\label{e311}\left(\int_{\mathbb{R}^{2}}|u|^{n} \right)^{1/n}\leq
\left(\int_{\mathbb{R}^{2}}[\,u^{2} + u_{x}^{2} +
(\partial_{x}^{-1}u_{y})^{2}\,]\right)^{1/2}.\end{aligned}$$ We will look at the most difficult terms to bound below. $$\begin{aligned}
\lefteqn{\left|\int
(\partial^{j}w_{xx})\,v_{xx}\,(\partial^{j}v_{xxx}) \right|}\\
& \leq &
\left(\int(\partial^{j}w_{xx})^{4} \right)^{1/4}
\left(\int(v_{xx})^{4} \right)^{1/4}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
& \leq & \left(\int[\,(\partial^{j}\,w_{xx})^{2} +
(\partial^{j}\,w_{xxx})^{2} +
(\partial^{j}\,w_{xy})^{2}\,] \right)^{1/2}\\
&  & \times \left(\int[\,(v_{xx})^{2} +
(v_{xxx})^{2} + (v_{xy})^{2}\,] \right)^{1/2}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
& \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2},\end{aligned}$$ while, $$\begin{aligned}
\lefteqn{\left|\int w_{xx}\,(\partial^{j}v_{xx})
\,(\partial^{j}v_{xxx}) \right|}\\
& = &
c\left|\int w_{xxx}\,(\partial^{j}v_{xx})^{2} \right|\\
& \leq &
c\left(\int w_{xxx}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{xx})^{4} \right)^{1/2}\\
& \leq & c\;||w||_{{Z}_{t}^{0}}\left(\int
[\,(\partial^{j}v_{xx})^{2} + (\partial^{j}v_{xxx})^{2} +
(\partial^{j}v_{xy})^{2}\,] \right)^{1/2}\\
& \leq & c\;||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{3},$ $$\begin{aligned}
\lefteqn{\left|\int
\partial^{j}(w_{x}\,v_{xxx})\,(\partial^{j}v_{xxx}) \right|}\\
& = & \left|\int[\,(\partial^{j}w_{x})\,v_{xxx}
+\,\ldots\,+
w_{x}\,(\partial^{j}v_{xxx})\,]\,(\partial^{j}v_{xxx}) \right|\\
& \leq & ||\partial^{j}w_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xxx}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
&  & +\,\ldots\,+ ||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}v_{xxx})^{2} \right)^{1/2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}
+\,\ldots\,+ c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Lastly, for $I_{4},$ $$\begin{aligned}
\left|\int\partial^{j}(w\,v_{xxxx})
\,(\partial^{j}v_{xxx}) \right| =
\left|\int[\,(\partial^{j}w)\,v_{xxxx} + \,\ldots\,
+
w\,(\partial^{j}v_{xxxx})\,]\,(\partial^{j}v_{xxx}) \right|.\end{aligned}$$ The first term is handled below. If $j=(0,\,0),$ then $$\begin{aligned}
\left|\int(\partial^{j}w)
\,v_{xxxx}\,(\partial^{j}v_{xxx}) \right| & = &
\left|\int w\,v_{xxxx}\,v_{xxx} \right|\\
& = & c\left|\int w_{x}\,v_{xxx}^{2} \right|\\
& \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xxx}^{2} \right)\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{0}}^{2},\end{aligned}$$ while, if $|j|>0,$ then $$\begin{aligned}
\left|\int(\partial^{j}w)\,w_{xxxx}
\,(\partial^{j}v_{xxx}) \right| & \leq &
||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xxxx}^{2}\right)^{1/2}
\left(\int(\partial^{j}v_{xxx})^{2}\right)^{1/2}\\
& \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ The last term in $I_{4}$ is handled below $$\begin{aligned}
\left|\int w\,(\partial^{j}v_{xxxx})
\,(\partial^{j}v_{xxx}) \right| & = &
c\left|\int w_{x}\,(\partial^{j}v_{xxx})^{2} \right|\\
& \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}v_{xxx})^{2} \right)\\
& \leq & c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Consequently, we conclude $$\begin{aligned}
\partial_{t}\int(\partial^{j}v_{xxx})^{2} \leq
c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next we take two $y$ derivatives of , multiply by $2\,(\partial^{j}v_{yy}),$ and integrate over $\mathbb{R}^{2}.$ Therefore, we have $$\begin{aligned}
\partial_{t}\int(\partial^{j}v_{yy})^{2} 
& \leq c\,\left|\int\partial^{j}(w\,v_{x})_{yy}
\,(\partial^{j}v_{yy}) \right|\\
& \leq c\,\left|\int\partial^{j}(w_{yy}\,v_{x} +
2\,w_{y}\,v_{xy} + w\,v_{xyy})
\,(\partial^{j}v_{yy}) \right|\\
& \leq c\,\left|\int\partial^{j}(w_{yy}\,v_{x})\,(\partial^{j}v_{yy}) \right|
+ c\left|\int\partial^{j}(w_{y}\,v_{xy})\,(\partial^{j}v_{yy}) \right| \\
& \qquad + c\left|\int\partial^{j}(w\,v_{xyy}) \,(\partial^{j}v_{yy}) \right|\\
& \leq I_{5} + I_{6} + I_{7}.\end{aligned}$$ First, we look at $I_{5},$ $$\begin{aligned}
\lefteqn{\left|\int\partial^{j}(w_{yy}\,v_{x})
\,(\partial^{j}v_{yy}) \right|}\\
& = & \left|\int[\,(\partial^{j}w_{yy})\,v_{x} +
\,\ldots\, +
w_{yy}\,(\partial^{j}v_{x})\,]\,(\partial^{j}v_{yy}) \right|\\
& \leq & c\,||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}w_{yy})^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\
&  & +\,\ldots\,+
c\,||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int w_{yy}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\
& \leq & c\,||v||_{{Z}_{t}^{0}}\,||w||_{{Z}_{t}^{|j|}}\,||v||_{Z_{t}^{|j|}}
+\,\ldots\,+ c\,||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ For $I_{6},$ $$\begin{aligned}
\lefteqn{\left|\int\partial^{j}(w_{y}\,v_{xy})
\,(\partial^{j}v_{yy}) \right|}\\
& = & \left|\int[\,(\partial^{j}w_{y})\,v_{xy} +
\,\ldots\, +
w_{y}\,(\partial^{j}v_{xy})\,]\,(\partial^{j}v_{yy}) \right|\\
& \leq & c\,||w_{y}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}v_{xy})^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\
&  & +\,\ldots\,+
c\,||\partial^{j}w_{y}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xy}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{yy})^{2} \right)^{1/2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}
+\,\ldots\,+ c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{0}}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Lastly, for $I_{7},$ $$\begin{aligned}
\left|\int\partial^{j}(w\,v_{xyy})\,(\partial^{j}v_{yy}) \right|
= \left|\int[\,(\partial^{j}w)\,v_{xyy} +
\,\ldots\,+
w\,(\partial^{j}v_{xyy})\,]\,(\partial^{j}v_{yy}) \right|.\end{aligned}$$ We will look at the first and last of these terms below. The rest of these terms are handled similarly. For the first term, if $j=(0,\,0),$ then we have $$\begin{aligned}
\left|\int(\partial^{j}w)
\,v_{xyy}\,(\partial^{j}v_{yy}) \right| & = &
\left|\int w\,v_{xyy}\,v_{yy} \right|\\
& = & c\left|\int w_{x}\,v_{yy}^{2} \right|\\
& \leq & c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\int v_{yy}^{2} \\
& \leq & c\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}^{2},\end{aligned}$$ while for $|j|>0,$ $$\begin{aligned}
\left|\int
(\partial^{j}w)\,v_{xyy}\,(\partial^{j}v_{yy}) \right| & \leq
& ||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xyy}^{2}\right)^{1/2}
\left(\int(\partial^{j}v_{yy})^{2}\right)^{1/2}\\
& \leq & ||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ The last term for $I_{7}$ is bounded as follows, $$\begin{aligned}
\left|\int w\,(\partial^{j}v_{xyy})\,(\partial^{j}v_{yy}) \right|
& = &
c\left|\int w_{x}\,(\partial^{j}v_{yy})^{2} \right|\\
& \leq &
c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int(\partial^{j}v_{yy})^{2} \\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Now apply one $t$ derivative to , multiply by $2\,(\partial^{j}v_{t})$ and integrate over $\mathbb{R}^{2}.$ We arrive at the following inequality, $$\begin{aligned}
\partial_{t}\int(\partial^{j}v_{t})  & \leq
& c\left|\int(\partial^{j}(w\,v_{x})_{t})
\,(\partial^{j}v_{t}) \right|\\
& \leq &
c\left|\int\partial^{j}(w_{t}\,v_{x})\,(\partial^{j}v_{t}) \right|
+
c\left|\int\partial^{j}(w\,v_{xt})\,(\partial^{j}v_{t})\,dx\,dt\right|\\
& = & I_{9} + I_{10}.\end{aligned}$$ For $I_{9},$ we have $$\begin{aligned}
\lefteqn{\left|\int\partial^{j}(w_{t}\,v_{x})
\,(\partial^{j}v_{t}) \right|}\\
& \leq &
c\left|\int(\partial^{j}w_{t})\,v_{x}\,(\partial^{j}v_{t}) \right|
+\,\ldots\,+
c\left|\int w_{t}\,(\partial^{j}v_{x})\,(\partial^{j}v_{t}) \right|\\
& \leq & c\,||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(\partial^{j}w_{t})^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\
&  & +\,\ldots\,+ ||\partial^{j}v_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int(w_{t})^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\
& \leq & c\,||v||_{{Z}_{t}^{0}}\,||w||_{Z_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}
+\,\ldots\,+
c\,||v||_{{Z}_{t}^{|j|}}\,||w||_{{Z}_{t}^{0}}\,||v||_{{Z}_{t}^{|j|}}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Next we look at $I_{10}.$ If $j=(0,\,0),$ we have $$\begin{aligned}
\left|\int w\,v_{xt}\,v_{t} \right| & = &
\left|\int w_{x}\,v_{t}^{2} \right|\\
& \leq &
||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int v_{t}^{2} \\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ If $j\neq (0,\,0),$ we have $$\begin{aligned}
\left|\int\partial^{j}(w\,v_{xt})\,(\partial^{j}v_{t}) \right|
& =
c\left|\int(\partial^{j}w)\,v_{xt}\,(\partial^{j}v_{t}) \right|
+\,\ldots \\
& \qquad +
c\left|\int w\,(\partial^{j}v_{xt})\,(\partial^{j}v_{t}) \right|\\
& = I_{10}(a) +\,\ldots\,+ I_{10}(\widetilde{a}).\end{aligned}$$ Now for $I_{10}(a),$ we use the following estimate $$\begin{aligned}
\left|\int(\partial^{j}w)\,v_{xt}\,(\partial^{j}v_{t}) \right|
& \leq & c\,||\partial^{j}w||_{L^{\infty}(\mathbb{R}^{2})}
\left(\int v_{xt}^{2} \right)^{1/2}
\left(\int(\partial^{j}v_{t})^{2} \right)^{1/2}\\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ While for $I_{10}(\widetilde{a}),$ we use the following estimate $$\begin{aligned}
\left|\int w\,(\partial^{j}v_{xt})\,(\partial^{j}v_{t}) \right|
& = &
c\left|\int w_{x}\,(\partial^{j}v_{t})^{2} \right|\\
& \leq &
c\,||w_{x}||_{L^{\infty}(\mathbb{R}^{2})}\int(\partial^{j}v_{t})^{2} \\
& \leq & c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Therefore, for $0\leq\widetilde{t}\leq t,$ we conclude that $$\begin{aligned}
& \partial_{t}\int[\,(\partial^{j}v(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}
+ (\partial^{j}v_{xxx}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} +
(\partial^{j}v_{yy}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2} +
(\partial^{j}v_{t}(\,\cdot\,,\,\cdot\,,\,\widetilde{t}))^{2}\,] \\
& \qquad \leq
c\,||w||_{{Z}_{t}^{|j|}}\,||v||_{{Z}_{t}^{|j|}}^{2}.\end{aligned}$$ Integrating with respect to $t,$ we obtain $$\begin{aligned}
||v||_{Z_{t}^{|j|}}^{2}\leq ||v(\,\cdot\,,\,\cdot\,,\,0))||_{H^{(|j|
+ 3,\,|j| + 2)}(\mathbb{R}^{2})}^{2} +
||v(\,\cdot\,,\,\cdot\,,\,0))||_{H^{|j|}(\mathbb{R}^{2})}^{2} +
c\,t\,||w||_{Z_{t}^{|j|}}\,||v||_{Z_{t}^{|j|}}^{2},\end{aligned}$$ as desired. $\square$

Uniqueness and Existence of a local solution
============================================

In this section, we will prove that for $\phi\in
X^{N}(\mathbb{R}^{2})$ there exists a unique solution of - in $L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})),$ where the time $T$ depends only $||\phi||_{X^{0}(\mathbb{R}^{2})}.$ First we prove uniqueness of solutions.\
\
[**Theorem 4.1**]{} (Uniqueness). [*Let $\phi\in
X^{0}(\mathbb{R}^{2})$ and $0<T<+\infty.$ Then there is at most one solution of - in $L^{\infty}([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ with initial data $u(x,\,y,\,0)=\phi(x,\,y).$*]{}\
\
[*Proof.*]{} Assume that $u,\,$ $v\in
L^{\infty}([0,\,T]:\,X^{0}(\mathbb{R}^{2}))$ are two solutions of - with $u_{t},\,$ $v_{t}\in
L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})),$ so all integrations below are justified and with the same initial data, in fact, with $(u - v)(x,\,y,\,0)=0.$ Then $$\begin{aligned}
\label{e401}&  & (u - v)_{t} + (u - v)_{xxx} + (u - v)_{x} +
(u\,u_{x} - v\,v_{x}) -
\partial_{x}^{-1}(u - v)_{yy} =0.\end{aligned}$$ By , $$\begin{aligned}
\label{e402}&  & (u - v)_{t} + (u - v)_{xxx} + (u - v)_{x} + (u -
v)\,u_{x} + (u - v)_{x}\,v -
\partial_{x}^{-1}(u - v)_{yy} =0.\end{aligned}$$ Multiplying by $2\,(u - v)$ and integrating with respect to $(x,\,y)$ over $\mathbb{R}^{2},$ $$\begin{aligned}
\label{e403}&  & 2\int (u - v)\,(u -
v)_{t} + 2\int (u - v)\,(u -
v)_{xxx} +
2\int (u - v)\,(u - v)_{x}   \\
&  & +\;2\int (u - v)^{2}\,u_{x} +
2\int (u - v)\,(u - v)_{x}\,v  -
2\int (u - v)\,\partial_{x}^{-1}(u -
v)_{yy} =0.\nonumber\end{aligned}$$ Integrating by parts each term in we obtain $$\begin{aligned}
\partial_{t}\int (u - v)^{2} & = &
-\;2\int (u - v)^{2}\,u_{x} +
\int (u - v)^{2}\,v_{x} \nonumber \\
& \leq & c\left(||u_{x}||_{L^{\infty}(\mathbb{R}^{2})} +
||v_{x}||_{L^{\infty}(\mathbb{R}^{2})}\right)\int(u
- v)^{2} \nonumber \\
\label{e404}& \leq & c\left(\,||u||_{X^{0}(\mathbb{R}^{2})} +
||v||_{X^{0}(\mathbb{R}^{2})}\,\right)\int (u - v)^{2}\end{aligned}$$ Using Gronwall’s inequality and the fact that $(u - v)$ vanishes at $t=0,$ it follows that $u=v.$ This proves the uniqueness of the solution. $\square$\
\
Now we consider existence of solutions to -. Our plan is to show that for $\phi \in X^N(\mathbb R^2)$ there exists a solution $u \in L^\infty([0,T]:X^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{X^0}$. In order to prove this we must first prove a preliminary result by introducing the following function space. Let $$\begin{aligned}
\label{e301}Y^{N}(\mathbb{R}^{2}) =
\left\{u:\;u,\,u_{xxx},\,u_{yy},\,
\frac{\eta^{2}}{\xi}\,\widehat{u}\in H^{N}(\mathbb{R}^{2})\right\}\end{aligned}$$ with the accompanying norm $$\begin{aligned}
\label{e302}||u||_{Y^{N}(\mathbb{R}^{2})}^{2}=\int_{\mathbb{R}^{2}}\left(u^{2}
+ \sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial_{x}^{-1}\partial^{j}u_{yy})^{2} +
(\partial^{j}u_{yy})^{2}\,]\right)dx\,dy\end{aligned}$$ where $j=(\alpha_1,\,\alpha_2)$ and $|j| = \alpha_1 + \alpha_2.$ We will begin by showing that, for $\phi \in Y^N(\mathbb R^2)$, there exists a solution $u$ of such that $u \in L^\infty([0,T]: Y^N(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{Y^0}$. Then we will prove a differential inequality of the form $$\partial_t \left(\int u^2 + u_{xxx}^2 + (\partial_x^{-1}u_{yy})^2\right) \leq
\left(\int u^2 + u_{xxx}^2 + (\partial_x^{-1}u_{yy})^2\right)^{3/2},$$ to show that in fact the solution $u$ obtained in Theorem 4.2 is in $L^\infty([0,T'];
X^0(\mathbb R^2))$ for a time $T'$ depending only on $||\phi||_{X^0(\mathbb R^2)}$. With these ideas in mind we state our existence theorem.\
(Existence). [*Let $k_{0}>0$ and $N$ be an integer $\geq 0.$ Then there exists a time $0<T<+\infty$ depending only on $k_{0}$ such that for all $\phi\in Y^{N}(\mathbb{R}^{2})$ with $||\phi||_{Y^{0}(\mathbb{R}^{2})}\leq k_{0}$ there exists a solution of , $u\in
L^{\infty}\left([0,\,T]:\,Y^{N}(\mathbb{R}^{2})\right)$ such that $u(x,\,y,\,0)=\phi(x,\,y).$*]{}\
\
The method of proof is as follows. As discussed in section 3, we begin by approximating by the linear equation . We construct the mapping $$\begin{aligned}
\Pi : Z_{T}^{N} \rightarrow Z_{T}^{N}\end{aligned}$$ where the initial condition is given by $u^{(n)}(x,\,y,\,0)=\phi(x,\,y)$ and the first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. Subsequent approximations are given by $u^{(n)} = \Pi(u^{(n-1)})$ for $n \geq 1$. Equation is a linear equation which by Lemma 3.1 can be solved at each iteration. We show that the sequence of solutions $\{u^{(n)}\}$ to our linear equation is bounded in $L^\infty([0,T];Y^0(\mathbb R^2))$ for a time $T$ depending only on $||\phi||_{Y^0}$. We then show that there is a subsequence of solutions to our approximate equations which converges to a solution $u \in L^\infty([0,T];
Y^0(\mathbb R^2))$ of . Lastly, we show that if $\phi \in Y^N(\mathbb R^2)$ for $N > 0$, then our solution $u \in L^\infty([0,T];Y^N(\mathbb R^2))$ where the time $T$ depends only on $||\phi||_{Y^0}$.\
\
It suffices to prove this result for $\phi\in\bigcap_{N\geq 0}H^{N}(\mathbb{R}^{2})$ and $\partial_{x}^{-1}\phi_{yy}\in\bigcap_{N\geq
0}H^{N}(\mathbb{R}^{2}).$ We can then use the same approximation procedure as before to prove the result for general initial data. Let $u^{(n)}$ be a solution of with initial data $u^{(n)}(x,y,0) =
\phi(x,y)$ and where the first approximation is given by $u^{(0)}(x,y,t) = \phi(x,y)$. By Lemma 3.2, we know that $$\begin{aligned}
\label{e405}||u^{(n)}||_{{Z}_{t}^{0}}^{2} \leq
||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2}
+
||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2}
+ c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{0}}\;||u^{(n)}||_{{Z}_{t}^{0}}^{2}.\end{aligned}$$ Further, using the fact that $||\phi||_{Y^{0}}\leq k_{0}$, we have $$\begin{aligned}
\lefteqn{||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2}
+
||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2}}
\\
& = &
||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2}\\
&  & +
\int [\,u_{xxx}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) +
u_{x}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) -
\partial_{x}^{-1}u_{yy}^{(n)}(\,\cdot\,,\,\cdot\,,\,0) +
u^{(n -
1)}(\,\cdot\,,\,\cdot\,,\,0)u_{x}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)\,]^{2} \\
& \leq & ||\phi||_{Y^{0}(\mathbb{R}^{2})}^{2} +
c\int [\,\phi_{xxx}^{2} + \phi_{x}^{2} +
(\partial_{x}^{-1}\phi_{yy})^{2} + (\phi\,\phi_{x})^{2}\,]
\\
& \leq & C\,||\phi||_{Y^{0}(\mathbb{R}^{2})}^{2} \leq C\,k_{0}^{2},\end{aligned}$$ where $\;C\,$ is independent of $n.\;$ Define $c_{0}=\left(\frac{C}{2}k_{0}^{2} + 1\right).\;$ Let $T_{0}^{(n)}$ be the maximum time such that $||u^{(j)}||_{Z_{t}^{0}}\leq c_{0}$ for $0\leq t\leq T_{0}^{(n)},$ $0\leq j\leq n.$ That is $$\begin{aligned}
T_{0}^{(n)}=\sup\{t\in [0,\,T_{0}^{(n)}]:\;||u^{(j)}||_{{Z}_{t}^{0}}\leq c_{0}\quad\mbox{for}\quad 0\leq j\leq n\}.\end{aligned}$$ Therefore, $$\begin{aligned}
\label{e406}||u^{(n)}||_{{Z}_{t}^{0}}^{2} & \leq &
||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(3,\,2)}(\mathbb{R}^{2})}^{2}
+
||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{L^{2}(\mathbb{R}^{2})}^{2}
+ c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{0}}\;||u^{(n)}||_{{Z}_{t}^{0}}^{2} \nonumber \\
& \leq & C\,k_{0}^{2} + c\,t\,c_{0}^{3}.\end{aligned}$$ [*Claim:*]{} $T_{0}^{(n)}$ does not approach $0.$\
\
On the contrary, assume that $T_{0}^{(n)}\rightarrow 0.$ Since $||u^{(n)}(\,\cdot\,,\,\cdot\,,\,t)||_{{Z}_{t}^{0}}$ is continuous for $t\geq 0,$ there exists $\tau\in [0,\,T]$ such that $c_{0}=||u^{(j)}(\,\cdot\,,\,\cdot\,,\,\tau)||_{{Z}_{\tau}^{0}}$ for $0\leq\tau\leq T_{0}^{(n)},$ $0\leq j\leq n.$ Then, by we have $$\begin{aligned}
\label{e407}c_{0}^{2} \leq C\,k_{0}^{2} +
c\,T_{0}^{(n)}\,c_{0}^{3}.\end{aligned}$$ As $n\rightarrow \infty,$ we have $$\begin{aligned}
\label{e408}\left(\frac{C}{2}\,k_{0}^{2} + 1\right)^{2} \leq
C\,k_{0}^{2}\quad \Longrightarrow \quad \frac{C^{2}}{4}\,k_{0}^{4}
+ 1 \leq 0\end{aligned}$$ which is a contradiction. Consequently $T_{0}^{(n)}\not\rightarrow
0.$ Choosing $T=T(c_0)$ sufficiently small, and $T$ not depending on $n,$ one concludes that $$\begin{aligned}
\label{e409}||u^{(n)}||_{{Z}_{t}^{0}}^{2} \leq
c\quad\mbox{for}\quad 0\leq t\leq T.\end{aligned}$$ This show that $T_{0}^{(n)}\geq T.$ Hence from we see that there exists a bounded sequence of solutions $u^{(n)}\in {Z}_{T}^{0}$ and therefore a subsequence $u^{(n_{j})}\equiv u^{(n)}$ such that $$\begin{aligned}
u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad
L^{\infty}([0,\,T]:\,H^{(3,\,2)}(\mathbb{R}^{2}))\end{aligned}$$ $$\begin{aligned}
u_{t}^{(n)}\stackrel{*}\rightharpoonup u_{t}\quad \mbox{weakly
in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Therefore, by Lions-Aubin’s compactness theorem there is a subsequence $u^{(n_{j})}\equiv u^{(n)}$ such that $u^{(n)}\rightarrow u$ strongly on $L^{\infty}([0,\,T]:\,H_{loc}^{1}(\mathbb{R}^{2})).$ Now it remains to show that each term in converges to its correct limit. First, $u_{xxx}^{(n)}\stackrel{*}\rightharpoonup
u_{xxx}$ weakly on $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Similarly $u_{t}^{(n)}\rightarrow u_{t}$ and $u_{x}^{(n)}\rightarrow u_{x}$ weak$^{*}$ in $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Now we will show that the nonlinear term converges to its correct limit. First, $u^{(n - 1)}\rightarrow u$ strongly in $L^{\infty}([0,\,T]:\,H_{loc}^{1}(\mathbb{R}^{2})).$ Moreover, $u_{x}^{(n - 1)}\stackrel{*}\rightharpoonup $ weakly in $L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ Therefore, $$\begin{aligned}
u^{(n - 1)}\,u_{x}^{(n)}\stackrel{*}\rightharpoonup u\,u_{x}\quad
\mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Consequently, $$\begin{split}
\partial_{x}^{-1}u_{yy}^{(n)}
& = u_{t}^{(n)} + u_{xxx}^{(n)} + u_{x}^{(n)} +
u^{(n)}\,u_{x}^{(n)}\\
&  \stackrel{*}\rightharpoonup u_{t} + u_{xxx} +
u_{x} + u\,u_{x}\quad \mbox{weakly in} \quad
L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).
\end{split}$$ But, also note that $$\begin{aligned}
u_{yy}^{(n)}\stackrel{*}\rightharpoonup u_{yy}\quad \mbox{weakly
in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).\end{aligned}$$ Therefore $$\begin{aligned}
\partial_{x}^{-1}u_{yy}^{(n)}
\stackrel{*}\rightharpoonup \partial_{x}^{-1}u_{yy}\quad
\mbox{weakly in}\quad L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2}))\end{aligned}$$ and consequently $u$ is a solution to . Now, we prove that there exists a solution to with $u\in
L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2}))$ for the time $T$ chosen above. We already know that there is a solution $u\in
L^{\infty}([0,\,T]:\,Y^{0}(\mathbb{R}^{2})).$ Therefore, it suffices to show that the approximating sequence $u^{(n)}$ is bounded in ${Z}_{T}^{N}$ and thus, by the convergence arguments above, our solution $u$ is in $L^{\infty}([0,\,T]:\;Y^{N}(\mathbb{R}^{2})).$ Again, by Lemma 3.1, we know our linearized equation can be solved in any interval of time in which the coefficients are defined. Therefore, for each iterate, $||u^{(n)}||_{{Z}_{t}^{N}}$ is continuous in $t\in
[0,\,T].$ By Lemma 3.2, it follows that $$\begin{aligned}
||u^{(n)}||_{{Z}_{t}^{N}}^{2}  & \leq &
||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N +
2)}(\mathbb{R}^{2})}^{2} \nonumber \\
\label{e410}&  & +\;
||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2}
+ c\,t\,||u^{(n - 1)}||_{{Z}_{t}^{N}}\;||u^{(n)}||_{{Z}_{t}^{N}}^{2}.\end{aligned}$$ On the other hand, as before and using $||\phi||_{Y^{N}}\leq
k_{N}$ we obtain $$\begin{aligned}
||u^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{(N + 3,\,N +
2)}(\mathbb{R}^{2})}^{2} +
||u_{t}^{(n)}(\,\cdot\,,\,\cdot\,,\,0)||_{H^{N}(\mathbb{R}^{2})}^{2}
&\leq  C\,k_{N}^{2},\end{aligned}$$ where $k_{N}$ is independent of $n.$ Define $c_{N}=\left(\frac{C}{2}\,k_{N}^{2} + 1\right).$ Let $T_{N}^{(n)}$ be the largest time that $||u^{(j)}||_{{Z}_{t}^{N}}\leq c_{N}$ for $0\leq t\leq T_{N}^{(n)},$ $0\leq j\leq n.$ That is, $$\begin{aligned}
T_{N}^{(n)}=\sup\{t\in [0,\,T_{N}^{(n)}]:\;||u^{(j)}||_{{Z}_{t}^{N}}\leq c_{N}\quad\mbox{for}\quad 0\leq j\leq n\}.\end{aligned}$$ Therefore, for $0\leq t\leq T_{N}^{(n)},$ $$\begin{aligned}
\label{e411}||u^{(n)}||_{{Z}_{t}^{N}}^{2}\leq C\,k_{N}^{2} +
c\,t\,c_{N}^{3}.\end{aligned}$$ [*Claim:*]{} $T_{N}^{(n)}$ does not approach $0.$\
\
On the contrary, assume that $T_{N}^{(n)}\rightarrow 0.$ Since $||u^{(n)}(\,\cdot\,,\,\cdot\,,\,t)||_{{Z}_{t}^{N}}$ is continuous for $t\geq 0,$ there exists $\tau\in [0,\,T]$ such that $c_{N}=||u^{(j)}(\,\cdot\,,\,\cdot\,,\,\tau)||_{{Z}_{\tau}^{N}}$ for $0\leq\tau\leq T_{N}^{(n)},$ $0\leq j\leq n.$ Then, by we have $$\begin{aligned}
\label{e412}c_{N}^{2} \leq C\,k_{N}^{2} +
c\,T_{N}^{(n)}\,c_{N}^{3}.\end{aligned}$$ As $n\rightarrow \infty,$ we have $$\begin{aligned}
\label{e413}\left(\frac{C}{2}\,k_{N}^{2} + 1\right)^{2} \leq
C\,k_{N}^{2}\quad \Longrightarrow \quad \frac{C^{2}}{4}\,k_{N}^{4}
+ 1 \leq 0\end{aligned}$$ which is a contradiction. Consequently $T_{N}^{(n)}\not\rightarrow
0.$ Choosing $T_N$ sufficiently small, and $T_N$ not depending on $n,$ one concludes that $$\begin{aligned}
\label{e414}||u^{(n)}||_{{Z}_{t}^{N}}^{2} \leq
c\quad\mbox{for}\quad 0\leq t\leq T_N.\end{aligned}$$ This show that $T_{N}^{(n)}\geq T_N.$ Now, let $$\begin{aligned}
T_{N}^{*} =\sup\{t\in [0,\,T_{N}^{*}]:\;u\in {Z}_{t}^{N}\}.\end{aligned}$$ We claim that $T_{N}^{*}\geq T$ and therefore, a time of existence can be chosen depending only on $||\phi||_{Y^{0}}.$ By Lemma 3.1 the linear equation can be solved in any interval of time in which the coefficients are defined, and thus $T_{N}^{*}\geq T.$ $\square$\
\
Now we want to improve our existence theorem. In particular, we want to show that the solution $u\in
L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2}))$ found in Theorem 4.2 is in $L^{\infty}([0,\,T']:\,X^{N}(\mathbb{R}^{2}))$ for a time $T'$ depending only on $||\phi||_{X^{0}(\mathbb{R}^{2})}.$ In order to do so, we first prove a differential inequality.\
\
[**Lemma 4.3.**]{} [*Let $u$ be the solution to our main equation in $L^{\infty}([0,\,T]:\,Y^{N}(\mathbb{R}^{2})).$ Then for any $0\leq t\leq T,$ we have*]{} $$\begin{aligned}
\lefteqn{\partial_{t}\int_{\mathbb{R}^{2}}\left(u^{2} +
\sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy}\nonumber
\\
\label{e415}& \leq & c\left[\int_{\mathbb{R}^{2}}\left(u^{2} +
\sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy\right]^{3/2}.\end{aligned}$$ [*Proof.*]{} We use a priori estimates on smooth solutions $u.$ Multiplying by $u$ and integrating over $\mathbb{R}^{2},$ it is straightforward to see that the $L^{2}(\mathbb{R}^{2})$-norm is conserved. Therefore, we only need to show that $$\begin{aligned}
\lefteqn{\partial_{t}\int \sum_{|j|\leq
N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]}\nonumber
\\
\label{e416}& \leq & c\left[\int \left(u^{2} +
\sum_{|j|\leq N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right) \right]^{3/2}.\end{aligned}$$ We consider the case $j=(0,\,0).$ The case $j\neq (0,\,0)$ is handled in a similar way.\
\
Applying $\partial_{x}^{3}$ to we obtain $$\begin{aligned}
\label{e417}&  & u_{xxxt} + u_{xxxxxx} + u_{xxxx} +
(u\,u_{x})_{xxx} - u_{xxyy} = 0.\end{aligned}$$ Multiplying by $2\,u_{xxx}$ and integrating over $\mathbb{R}^{2}$ we obtain $$\begin{aligned}
&  & 2\int u_{xxx}\,u_{xxxt} +
2\int u_{xxx}\,u_{xxxxxx} +
2\int u_{xxx}\,u_{xxxx} \nonumber \\
\label{e418}&  &
+\;2\int u_{xxx}\,(u\,u_{x})_{xxx} -
2\int u_{xxx}\,u_{xxyy} = 0.\end{aligned}$$ Using in straightforward integration by parts, we obtain $$\begin{aligned}
\lefteqn{\partial_{t}\int u_{xxx}^{2} = -\;2\int
u_{xxx}\,(u\,u_{x})_{xxx}} \\
& = & -\;2\int [\,3\,u_{xx}^{2} + 4\,u_{x}\,u_{xxx} +
u\,u_{xxxx}\,]\,u_{xxx}
 \\
& = & -\;7\int u_{x}\,u_{xxx}^{2}
 \leq 7\,||u_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\int_{\mathbb{R}^{2}}u_{xxx}^{2} \\
& \leq & c\left(\int [\,u_{x}^{2} + u_{xxx}^{2} +
u_{xy}^{2}\,]\right)^{1/2}\int_{\mathbb{R}^{2}}u_{xxx}^{2}
\\
& \leq & c\left(\int [\,u^{2} + u_{xxx}^{2} +
(\partial_{x}^{-1}u_{yy}^{2})^{2}\,] \right)^{3/2}.\end{aligned}$$ In a similar way, but now apply $\partial_{x}^{-1}\partial_{y}^{2}$ to instead of $\partial_{x}^{3}$ and multiply by $2\,\partial_{x}^{-1}u_{yy}$ instead of $2\,u_{xxx}$ we get $$\begin{aligned}
\lefteqn{\partial_{t}\int (\partial_{x}^{-1}u_{yy})^{2}}\\
& \leq &
c\left|\int (u^{2})_{yy}\,(\partial_{x}^{-1}u_{yy})\right|
= c\left|\int [\,u_{yy}\,u +
u_{y}^{2}\,]\,(\partial_{x}^{-1}u_{yy})\right|\\
& \leq &
c\left|\int u_{x}\,(\partial_{x}^{-1}u_{yy})^{2}\right|
+
\left|\int u_{y}^{2}\,(\partial_{x}^{-1}u_{yy})\right|\\
& \leq & c||u_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\int (\partial_{x}^{-1}u_{yy})^{2} +
\left(\int u_{y}^{4} \right)^{1/2}
\left(\int (\partial_{x}^{-1}u_{yy})^{2} \right)^{1/2}\\
& \leq & c||u_{x}||_{L^{\infty}(\mathbb{R}^{2})}
\int (\partial_{x}^{-1}u_{yy})^{2} +
\left(\int [\,u_{y}^{2} + u_{xy}^{2} +
(\partial_{x}^{-1}u_{yy})^{2}\,] \right)
\left(\int (\partial_{x}^{-1}u_{yy})^{2} \right)^{1/2}\\
& \leq & c\left(\int [\,u^{2} + u_{xxx}^{2} +
(\partial_{x}^{-1}u_{yy})^{2}\,]\right)^{3/2}.\end{aligned}$$ The lemma follows. $\square$\
\
[**Corollary 4.4.**]{} [*Let $u$ be the solution to with initial data $\phi\in Y^{N}(\mathbb{R}^{2}).$ Denote by $0<T<+\infty$ the life span of this solution in $Y^{N}(\mathbb{R}^{2}).$ Then there exists $0<T'\leq T,$ depending only on the norm of $\phi\in X^{0}(\mathbb{R}^{2})$ such that $u\in L^{\infty}([0,\,T']:\,X^{N}(\mathbb{R}^{2})).$*]{}\
\
[*Proof.*]{} Let $$\begin{aligned}
h(t) = \int_{\mathbb{R}^{2}}\left(u^{2} + \sum_{|j|\leq
N}[\,(\partial^{j}u_{xxx})^{2} +
(\partial^{j}(\partial_{x}^{-1}u_{yy}))^{2}\,]\right)dx\,dy\equiv
||u||_{X^{N}}^2.\end{aligned}$$ Using we have $h'(t)\leq c\;[\,h(t)\,]^{3/2}.$ Integrating this inequality with respect to $t$, we obtain that $h(t)^{1/2}\leq c/(h(0)^{-1/2} - t)$ and therefore, we get a lower bound on the time of existence of $h(t)$ depending only on $h(0).$ $\square$\
\
[**Corollary 4.5.**]{} [*Let $\phi\in X^{N}(\mathbb{R}^{2})$ for some $N\geq 0$ and let $\phi^{(n)}$ be a sequence converging to $\phi$ in $X^{N}(\mathbb{R}^{2}).$ Let $u$ and $u^{(n)}$ be the corresponding unique solutions, given by Theorems 4.1 and 4.2 and Corollary 4.4, in $L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))$ for a time $T$ depending only on $\sup_{n}||\phi^{(n)}||_{X^{0}(\mathbb{R}^{2})}.$ Then* ]{} $$\begin{aligned}
\label{e419}u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly
in}\quad L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})).\end{aligned}$$ [*Proof.*]{} By assumption $u\in
L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})),$ then there exists a weak$^{*}$ convergent subsequence, still denoted $u^{(n)}$ such that $$\begin{aligned}
u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad
L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))\hookrightarrow
L^{\infty}([0,\,T]:\,H^{1}(\mathbb{R}^{2})).\end{aligned}$$ Moreover, by equation , $u^{(n)}\in
L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2}))$ implies $u_{t}^{(n)}\in L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})).$ By The Lions-Aubin compactness theorem, $$\begin{aligned}
u^{(n)}\rightarrow u\quad \mbox{strongly in}\quad
L^{\infty}([0,\,T]:\,H_{loc}^{1/2}(\mathbb{R}^{2})).\end{aligned}$$ Now we just need to show that each term in converges to its correct limit, and $u_{t}^{(n)}\rightarrow u_{t}$ for $u\in
L^{\infty}([0,\,T]:\,X^{N}(\mathbb{R}^{2})).$\
\
The only thing we need to show is that the nonlinear term converges to its correct limit, namely that $u^{(n)}\,u_{x}^{(n)}\rightarrow u\,u_{x}.$ We know that $u_{x}^{(n)}\stackrel{*}\rightharpoonup u_{x}$ weakly in $
L^{\infty}([0,\,T]:\,H^{1}(\mathbb{R}^{2}))$ and $u^{(n)}\rightarrow u$ strongly in $
L^{\infty}([0,\,T]:\,H_{loc}^{1/2}(\mathbb{R}^{2})).$ Therefore, their product converges in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2})).$ Clearly, the linear terms also converge in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2}))$ and therefore, we conclude that $u_{t}^{(n)}\rightarrow u_{t}$ in $L^{2}([0,\,T]:\,L_{loc}^{1}(\mathbb{R}^{2})).$ The proof follows. $\square$

Estimate of error terms
=======================

In this section we prove the main estimates used in our gain of regularity theorem.\
\
\
\
\
The idea of the proof is the following. For a given $\alpha$ satisfying the hypotheses above, we choose a weight function $f_\alpha \approx t^{|\alpha|-L}
x^{2L-|\alpha|-\alpha_2}$ for $x > 1$ and $f_\alpha \approx t^{|\alpha|-L}e^{\sigma x}$ for $x < -1$. Then with this choice of weight function, we apply the operator $\partial^\alpha$ to , multiply by $f_\alpha\pa u$ and integrate over $\mathbb R^2$ to obtain the main equality stated in . In this theorem, we bound the last three terms on the left-hand side of by terms of the form and .\
For each $\alpha$ we apply the operator $\partial^\alpha$ to (2.1), multiply our differentiated equation by $2 \fa (\pa u)$ where we take $$\fa(x,t) = \int_{-\infty}^x g_\alpha(z,t) \,dz \quad \text{for } g_\alpha \in
  W_{\sigma, 2L-|\alpha|-\alpha_2-1,|\alpha|-L},$$ and integrate over $\mathbb R^2 \times [0,t]$ for $0 \leq t \leq T$. As stated in Lemma 2.1, we arrive at our main equality $$\begin{aligned}
& \partial_t \int \fa (\pa u)^2 + 3 \int (\fa)_x (\pa u_x)^2 - \int [(\fa)_t + (\fa)_{xxx} +
(\fa)_x] (\pa u)^2 \\
& \qquad - \int (\fa)_x (\pa \partial_x^{-1}u_y)^2 + 2 \int \fa (\pa u)\pa(uu_x) = 0.\end{aligned}$$ Using (1.5) and $f(\cdot,0) = 0$ we get the following identity after integrating with respect to $t$, $$\label{before-estimating}
\begin{split}
& \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa u_x)^2 \\
& \leq \int_0^T \int (\fa)_x (\pa \partial_x^{-1} u_y)^2 + C \int_0^T \int \fa (\pa u)^2
 + 2 \left|\int_0^T \int \fa (\pa u) \pa (uu_x) \right|.
\end{split}$$ We notice that the first term on the right-hand side of can be written as $$\label{inverse-term}
\int_0^T \int (\fa)_x (\pa \partial_x^{-1} u_y)^2 = C \int_0^T \int t g_\gamma (\partial^\gamma u_x)^2$$ for some $g_\gamma \in W_{\sigma,2L-|\gamma|-\gamma_2-1,|\gamma|-L}$ where $\gamma = (\alpha_1-2,\alpha_2+1)$. Further, we notice that $2L-|\gamma|-\gamma_2 \geq 1$ and $\alpha_1 \geq 2$ since $2L-|\alpha|-\alpha_2\geq 1$ and $L+1 \leq |\alpha|$. Therefore, is of the form specified by . Therefore, $$\label{after-inverse-term}
\begin{split}
& \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa u_x)^2 \\
& \qquad \leq C + C \int_0^T \int \fa (\pa u)^2
 + 2 \left|\int_0^T \int \fa (\pa u) \pa (uu_x) \right|
\end{split}$$ where $C$ depends only on terms of the form . We now need to estimate the term $$\left|\int_0^T \int \fa (\pa u)\pa (uu_x)\right|.$$ Each term is of the form $$\left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right|.$$ where $r_1 + s_1 = \alpha_1$, $r_2 + s_2 = \alpha_2$. Below we consider all terms of level $|\alpha|$.

In this case, $|r|=0$, and we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T
  \int \fa (\pa u) u (\pa u_x)\right| \\
  & = C \left|\int_0^T \int (\fa u)_x (\pa u)^2\right| \\
  & \leq C ||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$ In this case, $|r|=1$ giving us the following two subcases:

\(a) [**The subcase $r = (1,0)$.**]{} In this case, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int
  \fa (\pa u)u_x (\pa u)\right| \\
  & \leq C||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$

\(b) [**The subcase $r = (0,1)$.**]{} We note that this case will only occur if $\alpha_2 \geq 1$. In this case, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T
  \int \fa (\pa u) u_y (\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u_x)\right| \\
  & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial_x^{\alpha_1+1}\partial_y^{\alpha_2-1}u)^2\right)^{1/2}\end{aligned}$$ Since $f_\alpha \approx x^{2L-|\alpha|-\alpha_2}$ as $x \rightarrow \infty$, it is clear that $f_\alpha \leq C f_{\alpha_1+1,\alpha_2-1}$

We have three subcases to consider.

\(a) [**The subcase $r = (2,0)$.**]{} In this case, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T \int
  \fa (\pa u)u_{xx} (\partial_x^{\alpha_1-2}\partial_y^{\alpha_2} u_x)\right| \\
  & \leq C||u_{xx}||_{L^\infty} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2} \\
  & \leq C ||u||_{X^1}\left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2}.\end{aligned}$$ The last term on the right-hand side above is of order $|\alpha|-1 \geq L$. The weight function $f_\alpha \in W_{\sigma, 2L-|\alpha|-\alpha_2,|\alpha|-L}$. Since $2L-|\alpha|-\alpha_2 < 2L-(\alpha_1-1+\alpha_2)
-\alpha_2$, we see that this term is bounded by a term of the form .

\(b) [**The subcase $r=(1,1)$.**]{} In this case, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T
  \int \fa (\pa u)u_{xy} (\partial_x^{\alpha_1-1}\partial_y^{\alpha_2-1}u_x)\right| \\
  & \leq C||u_{xy}||_{L^\infty} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial_x^{\alpha_1}\partial_y^{\alpha_2-1} u)^2\right)^{1/2} \\
  & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial_x^{\alpha_1}\partial_y^{\alpha_2-1} u)^2\right)^{1/2}\end{aligned}$$ Using the fact that $\fa \in W_{\sigma,2L-|\alpha|-\alpha_2,|\alpha|-L}$ and $2L-|\alpha|
-\alpha_2 < 2L-(\alpha_1+\alpha_2-1) - (\alpha_2-1)$, we conclude that the last term is bounded by a term of the form .

\(c) [**The subcase $r=(0,2)$.**]{} In this case, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & = \left|\int_0^T
  \int \fa (\pa u)u_{yy} (\partial_x^{\alpha_1} \partial_y^{\alpha_2-2}u_x)\right| \\
  & \leq \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int u_{yy}^4
  \right)^{1/4} \\
  & \qquad \times \left(\int_0^T \int (\fa^{1/2} \partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u)^4
  \right)^{1/4}.\end{aligned}$$ Now $$\left(\int_0^T \int u_{yy}^4\right)^{1/4} \leq C \left(\int_0^T \int [u_{yy}^2 + u_{xyy}^2 +
(\partial_x^{-1} u_{yyy})^2]\right)^{1/2} \leq C||u||_{X^1}.$$ Further, $$\begin{aligned}
  & \left(\int_0^T \int (\fa^{1/2} \partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u)^4 \right)^{1/4}\\
  & \leq C \left(\int_0^T \int \fa (\partial_x^{\alpha_1+1} \partial_y^{\alpha_2-2}u)^2
  + \fa (\partial_x^{\alpha_1+2}\partial_y^{\alpha_2-2}u)^2 + \fa (\partial_x^{\alpha_1}
  \partial_y^{\alpha_2-1}u)^2    \right)^{1/2}\end{aligned}$$ Now the first and third terms in the integrand are of order $|\alpha|-1$ and are clearly bounded by terms of the form . The second term in the integrand is of order $|\alpha|$. It will be bounded using Gronwall’s inequality (and using the fact that the order of the $y$ derivative is less than $\alpha_2$ and therefore this terms can handle an even greater power of $x$.)

In this case $|r|=3$. First, we consider the case in which $L \geq 4$. Since $|\alpha| \geq L+1$, we note that $|s|+2 = |\alpha|-1 \geq L$. Using this fact, we bound as follows. $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq \left(\int_0^T
  \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int (\partial^r u)^4\right)^{1/4} \\
  & \qquad \times 
  \left(\int_0^T \int (\fa^{1/2} \partial^s u_x)^4\right)^{1/4}\end{aligned}$$ Now $$\begin{aligned}
  \left(\int_0^T \int (\partial^r u)^4\right)^{1/4} \leq C\left(\int_0^T \int
  [(\partial^r u)^2 + (\partial^r u_x)^2 + (\partial^r u_y)^2]\right)^{1/2}.\end{aligned}$$ Since $|r| = 3$, each of these terms is at most of order $4 \leq L \leq |\alpha|-1$, and, therefore, bounded by terms of the form . Similarly, $$\label{s-alpha-3}
  \left(\int_0^T \int (\fa^{1/2} \partial^s u_x)^4\right)^{1/4} \leq C \left(\int_0^T \int
  \fa [(\partial^s u_x)^2 + (\partial^s u_{xx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}$$ We notice that the last two terms on the right-hand side of are of order $|s|+2=|\alpha|-1\geq L$. In order to verify that we have the correct power of $x$, we note that $$2L-|\alpha|-\alpha_2 \leq 2L-(|s|+2)-(s_2+1)$$ since $|s|= |\alpha|-3$. Therefore, we can conclude that each of those terms is bounded by a term of the form . Finally, we look at the first term on the right-hand side of . If $|s|+1 \geq L$, then this term is bounded by as the other two terms. If $|s|+1 < L$, then using the fact that $|s|+2 \geq L$, we conclude that $|s| = L-2$, and, therefore, $$\begin{aligned}
2L-|\alpha|-\alpha_2 & \leq 2L-(|s|+2)-(s_2+1) \\
&\leq s_1+1\end{aligned}$$ Therefore, we conclude that the first term above is bounded by a term of the form of order $|s|+1 < L$.

We now look at the cases when $L=2$ or $L=3$. In either case, if $|\alpha| \geq 5$, then we can handle as above. We first consider the case when $|\alpha|=4$ ($L=2$ or $L=3$). In this case, using the fact that $|r|=3$ and $|s|=1$, we have $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq
  C||\partial^s u_x||_{L^\infty} \left(\int_0^T
  \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T \int \fa (\partial^r u)^2 \right)^{1/2} \\
  & \leq C||u||_{X^1} \left(\int_0^T \int \fa (\pa u)^2\right)^{1/2} \left(\int_0^T
  \int \fa (\partial^r u)^2\right)^{1/2}.\end{aligned}$$ Since $|r|=3=|\alpha|-1$ and $2L-|\alpha|-\alpha_2 \leq 2L-|r|-r_2$, we see that the last term above is bounded by a term of the form .

Last, we consider $|\alpha|=3$. In this case, we must have $L=2$, $|r|=3$ and $|s|=0$. Therefore, $r = \alpha$. We bound as follows: $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x) \right| & \leq C||u_x||_{L^\infty}
  \int_0^T \int \fa (\pa u)^2 \\
  & \leq C||u||_{X^1} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$

We consider the set $A = \{x: x > 1\}$. The set $A_{-1} = \{x < -1\}$ can be handled similarly. We have $$\begin{aligned}
  & \left|\int_0^T \int_A \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| \\
  & \leq C T^M||
  t^{\nu_s}\partial^s u_x||_{L^\infty(A)} \left(\int_0^T \int_A \fa (\pa u)^2\right)^{1/2}
  \left(\int_0^T \int_A t^{\nu_r} x^{2L-|\alpha|-\alpha_2}(\partial^r u)^2\right)^{1/2}\end{aligned}$$ where $\nu_s = \frac{(|s|+3-L)^+}2$ and $\nu_r = \frac{(|r|-L)^+}2$. First, we must verify that $M \geq 0$. We see that $$\begin{aligned}
M & = \frac {|\alpha|-L}2 - \nu_s - \nu_r \\
& = \frac{|\alpha|-L-(|s|+3-L)^+-(|r|-L)^+}2 \\
& \geq \frac {|\alpha|-|s|-|r|-3+L}2 \\
& = \frac {L-3}2 \geq 0\end{aligned}$$ as long as $L \geq 3$. By assumption, $L \geq 2$. If $L = 2$, then $|\alpha| = 3$. In that case, we cannot have $|s| \leq |\alpha| - 4$. Therefore, we conclude that $M \geq 0$. Further, $$\begin{aligned}
  ||t^{\nu_s}\partial^s u_x||_{L^\infty} & \leq C \left(\int t^{2\nu_s} [(\partial^su_x)^2
  + (\partial^s u_{xxx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Each of those terms is of order at most $|\alpha|-1$, and therefore, bounded by terms of the form . Further, $2L-|\alpha|-\alpha_2 \leq 2L-|r|-r_2$. Therefore, for $|r| \leq
|\alpha|-1$, the last term above is bounded by terms of the form . In the case that $|r|=|\alpha|$, we have $|s| = 0$, and therefore $$\begin{aligned}
  \left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right| & \leq C ||u_x||
  _{L^\infty} \int_0^T \int \fa (\pa u)^2 \\
  & \leq C ||u||_{X^0} \int_0^T \int \fa (\pa u)^2.\end{aligned}$$

Combining our estimates above on $$\left|\int_0^T \int \fa (\pa u)(\partial^r u)(\partial^s u_x)\right|$$ with , we see that $$\begin{split}
\label{power-1} \int \fa (\pa u)^2 + 3 \int_0^T \int (\fa)_x (\pa
u_x)^2 & \leq C + C\sum_{\stackrel{|\gamma| = |\alpha|}{
2L-|\gamma|-\gamma_2 \geq 1}} \int_0^T \int f_\gamma
(\partial^\gamma u)^2
\end{split}$$ where the constant $C$ depends only on terms of the form and .

Using the above estimate for all derivatives $\gamma$ of order $|\alpha|$ such that $2L-|\gamma|-\gamma_2 \geq 1$, we see that $$\begin{split}
\sum_{\stackrel{|\gamma|=|\alpha|}{2L-|\gamma|-\gamma_2 \geq 1}}&
\left[\int f_\gamma (\pa u)^2 +
3 \int_0^T \int (f_\gamma)_x (\pa u_x)^2\right] \\
& \leq C + \sum_{\stackrel{|\gamma|=|\alpha|}{2L-|\gamma|-\gamma_2
\geq 1}}\int_0^T \int f_\gamma (\partial^\gamma u)^2.
\end{split}$$ where $C$ depends only on terms of the form and . Applying Gronwall’s inequality, we get the desired estimate. $\square$\
\

Persistence Theorem
===================

In section four we proved the existence of a solution $u$ to in $L^\infty([0,T];
X^N(\mathbb R^2))$ for given initial data $\phi \in X^N(\mathbb R^2)$. In this section, we prove that if, in addition, our initial data $\phi$ lies in the weighted space $\widetilde H_x^K(W_{0\;K\;0})$ for some $K \geq 0$, then the solution $u$ also lies in $L^\infty([0,T];\widetilde H_x^K(W_{0\;K\;0}))$. This property is known as a “persistence" property of the initial data. This property provides a basis for starting the induction in our Gain of Regularity theorem in Section 7.\
\

We use induction on $j=|\alpha|$ for $1 \leq j \leq K$. The case that $j = 0$ follows from conservation of $L^2$ norm. We derive formally some a priori estimates for the solution where the bound involves only the norms of $u \in L^\infty([0,T]:X^1(\mathbb R^2))$ and the norms of $\phi \in \widetilde H_x^K(
W_{0\;K\;0})$. Then, we can apply convergence arguments to show that the result holds true for general solutions. In order to do so, we need to approximate general solutions $u \in X^1(\mathbb R^2)$ by smooth solutions and approximate general weight functions $f \in W_{0\;j\;0}$ by smooth, bounded weight functions. The first of these procedures has already been discussed, so we will concentrate on the second.

For a fixed $i$, we begin by taking a sequence of bounded weight functions $g_{i,\delta}$ which decay as $|x| \rightarrow \infty$ and which approximate $g_i \in W_{\sigma\; i-1\; 0}$ with $\sigma > 0$ from below, uniformly on any half-line $(-\infty,c)$. Define the weight functions $$f_{i,\delta}(x,t) = 1 + \int_{-\infty}^x g_{i,\delta}(z,t) \,dz.$$ Therefore, the functions $f_{i,\delta}$ are bounded weight functions approximating $f_i \in W_{0,i,0}$ from below, uniformly on compact sets.

&gt;From (5.3) and using the fact that $\partial_t (\fid) \leq c \fid$ and $\partial_x (\fid) \leq c
\fid$, we have $$\begin{aligned}
  & \int \fid (\pa u)^2 + 3 \int_0^T \int (\fid)_x (\pa u_x)^2 \,dt
   \leq \int_0^T (\fid)_x (\pa \partial_x^{-1} u_y)^2 \,dt \\
   & \qquad + C\int_0^T
  \int \fid (\pa u)^2 \,dt + 2 \left|\int_0^T \int \fid (\pa u) \pa (uu_x) \,dt \right|.\end{aligned}$$

\(a) [**The subcase $\alpha = (1,0)$.**]{} Defining $g_{1,\delta}$ and $f_{1,\delta}$ as above, we see that $f_{1,\delta}$ will approximate $f_1 \in W_{0\;1\;0}$ from below. Differentiating (2.1) in the $x-$variable, multiplying by $2f_{1,\delta}$ and integrating over $\mathbb R^2$, we have $$\begin{split}
\label{alpha1beta0}
& \partial_t \int \foned u_x^2 + 3 \int(\foned)_xu_{xx}^2 \\
& \qquad = \int (\foned)_x u_y^2 + \int [\partial_t\foned + \partial_x^3\foned + \partial_x
\foned]u_x^2 - 2 \int \foned u_x(uu_x)_x \\
& \leq C \int u_y^2 + C \int \foned u_x^2 + 2 \left|\int \foned u_x(uu_x)_x\right|
\end{split}$$

Moreover $$\begin{aligned}
  \left|\int \foned u_x(uu_x)_x \right|
  & = \left|\int \foned u_x[u_x^2 + uu_{xx}]\right| \\
  & = \left|\int \foned [u_x^3 + uu_{x}u_{xx}] \right| \\
  & \leq c ||u_x||_{L^\infty} \int \foned u_x^2 + \frac 12\left|\int (\foned u)_x u_x^2 \right| \\
  & \leq c(||u||_{L^\infty} + ||u_x||_{L^\infty})\int \foned u_x^2 \\
  & \leq C ||u||_{X^0}\int \foned u_x^2.\end{aligned}$$ Combining this estimate with , we conclude that for $0 \leq t \leq T$, $$\int \foned(\cdot,t) u_x^2 + 3 \int_0^T \int (\foned)_x u_{xx}^2 \leq C \int \foned(\cdot,0)
\phi_x^2 + C \int_0^T \int u_y^2 + C \int_0^T \int \foned u_x^2.$$ Applying Gronwall’s inequality, we conclude that $$\sup_{0 \leq t \leq T} \int \foned u_x^2 + 3 \int_0^T \int (\foned)_x u_{xx}^2 \leq C$$ where $C$ does not depend on $\delta$ but only on $T$ and the norm of $\phi \in
X^1(\mathbb R^2) \cap \widetilde H_x^1(W_{0\;1\;0})$. Taking the limit as $\delta\rightarrow \infty$, we conclude that $$\sup_{0 \leq t \leq T} \int f_1 u_x^2 + 3 \int_0^T \int g_1 u_{xx}^2 \leq C,$$ as claimed.

\(b) [**The subcase $\alpha = (0,1)$.**]{} Here, our weight function $f_0 \in W_{0\;0\;0}$. Differentiating (2.1) in the $y$-variable, multiplying by $2\fzerod u_y$ and integrating over $\mathbb R^2$, we have $$2 \int \fzerod u_y u_{yt} + 2 \int \fzerod u_y u_{xxxy} - 2 \int \fzerod u_y \partial_x^{-1}
u_{yyy} + 2 \int \fzerod u_y u_{xy} + 2 \int \fzerod u_y (uu_x)_y = 0.$$ Integrating each term by parts gets $$\begin{split}
    \label{alpha0beta1}
    & \partial_t \int \fzerod u_y^2 + 3 \int (\fzerod)_x u_{xy}^2 \\
    & \qquad \leq \int (\fzerod)_x (\partial_x^{-1}u_{yy})^2 + C \int \fzerod u_y^2
    + 2 \left|\int \fzerod u_y (uu_x)_y\right| \\
    & \leq C \int (\partial_x^{-1} u_{yy})^2 + C \int \fzerod u_y^2 + 2 \left|
    \int \fzerod u_y (uu_x)_y\right|.
  \end{split}$$ Moreover, $$\begin{aligned}
  \left|\int \fzerod u_y (uu_x)_y\right| & = \left|\int \fzerod u_y (u_yu_x + uu_{xy})\right| \\
  & \leq C ||u_x||_{L^\infty} \int \fzerod u_y^2 + C ||u||_{L^\infty} \int (\fzerod)_x u_y^2 \\
  & \leq C ||u||_{X^0}\int \fzerod u_y^2.\end{aligned}$$ Combining this estimate with , we conclude that for $0 \leq t \leq T$, $$\int \fzerod(\cdot,t)u_y^2 + 3 \int_0^T \int (\fzerod)_x u_{xy}^2 \leq \int \fzerod(\cdot,0)
  \phi_y^2 + C \int_0^T \int (\partial_x^{-1}u_{yy})^2 + C \int_0^T \int \fzerod u_y^2.$$ Applying Gronwall’s inequality, we conclude that $$\sup_{0 \leq t \leq T} \int \fzerod u_y^2 + 3 \int_0^T \int (\fzerod)_x u_{xy}^2 \leq C$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^0(\mathbb R^2)
\cap \widetilde H_x^1(W_{0\;1\;0})$. Passing to the limit, we conclude that $$\sup_{0 \leq t \leq T} \int f_0 u_y^2 + 3 \int_0^T \int g_0 u_{xy}^2 \leq C.$$

\(a) [**The subcase $\alpha = (2,0)$.**]{} In this case, $\ftwod$ will approximate $f_2 \in W_{0\;2\;0}$. In a similar way as above, we have $$\begin{aligned}
  & 2 \int \ftwod u_{xx} u_{xxt} + 2 \int \ftwod u_{xx}u_{xxxxx} - 2 \int \ftwod u_{xx} u_{xyy} \\
  & \qquad + 2 \int \ftwod u_{xx}u_{xxx} + 2 \int \ftwod u_{xx}(uu_x)_{xx} = 0.\end{aligned}$$ Integrating each term by parts gets $$\begin{aligned}
  & \partial_t \int \ftwod u_{xx}^2 + 3 \int (\ftwod)_xu_{xxx}^2 \\
  & \qquad = c \int (\ftwod)_x u_{xy}^2 + \int [\partial_t \ftwod + \partial_x^3 \ftwod
  + \partial_x \ftwod]u_{xx}^2 - 2 \int \ftwod u_{xx} (uu_x)_{xx} \\
  & \leq \int \ftwod u_{xy}^2 + c \int \ftwod u_{xx}^2 + 2 \left|\int \ftwod u_{xx}
  (uu_x)_{xx} \right|.\end{aligned}$$ Moreover $$\begin{aligned}
  \left|\int \ftwod u_{xx}(uu_x)_{xx} \right|
  & = \left|\int \ftwod u_{xx}[3u_{x} u_{xx} + uu_{xxx}] \right| \\
  & = \left|\int \ftwod[3u_{x} u_{xx}^2 + uu_{xx} u_{xxx}] \right| \\
  & \leq c||u_x||_{L^\infty} \int \ftwod u_{xx}^2 + \frac{1}{2}\left|(\ftwod u)_x u_{xx}^2 \right| \\
  & \leq c(||u||_{L^\infty} + ||u_x||_{L^\infty})\int \ftwod u_{xx}^2 \\
  & \leq C ||u||_{X^0} \int \ftwod u_{xx}^2.\end{aligned}$$ Therefore, $$\partial_t \int \ftwod u_{xx}^2 + 3 \int (\ftwod)_x u_{xxx}^2 \leq \int (\ftwod)_x u_{xy}^2
+ C \int \ftwod u_{xx}^2,$$ where $C$ depends only on the norm of $\phi \in X^0(\mathbb R^2)$. We will combine this estimate with the estimate below.

\(b) [**The subcase $\alpha = (1,1)$**]{} Applying $\partial_x\partial_y$ to (2.1), multiplying by $2 \foned u_{xy}$ where $\foned$ approximates $f_1 \in W_{0\;1\;0}$, and integrating over $\mathbb R^2$, we have $$\begin{aligned}
& 2 \int \foned u_{xy} u_{xyt} + 2 \int \foned u_{xy}u_{xxxxy} - 2 \int \foned u_{xy}u_{yyy} \\
& \qquad + 2 \int \foned u_{xy}u_{xxy} + 2 \int \foned u_{xy}(uu_x)_{xy} = 0.\end{aligned}$$ Integrating each term by parts gets $$\begin{aligned}
  & \partial_t \int \foned u_{xy}^2 + 3 \int (\foned)_x u_{xxy}^2 \\
  & \qquad = \int (\foned)_x u_{yy}^2 + \int [\partial_t \foned + \partial_x^3 \foned +
  \partial_x \foned]u_{xy}^2 - 2 \int \foned u_{xy}(uu_x)_{xy} \\
  & \leq \int (\foned)_x u_{yy}^2 + c \int \foned u_{xy}^2 + 2 \left|\int \foned u_{xy}
  (uu_x)_{xy}\right|.\end{aligned}$$ Moreover, $$\begin{aligned}
  \left|\int \foned u_{xy}(uu_x)_{xy}\right|
  & = \left|\int \foned u_{xy}(2u_x u_{xy} + u_{xx}u_y + uu_{xxy})\right| \\
  & \leq C(||u_x||_{L^\infty} + ||u||_{L^\infty})\int \foned u_{xy}^2 +
  \left|\int \foned u_{xx}u_{xy} u_y\right| \\
  & \leq C||u||_{X^0}\int \foned u_{xy}^2 + C||u_y||_{L^\infty} (\int \foned u_{xx}^2
  + \int \foned u_{xy}^2) \\
  & \leq C||u||_{X^0} \int \foned u_{xy}^2 + C ||u||_{X^1} \int \foned (u_{xx}^2 + u_{xy}^2)\end{aligned}$$ since $$\begin{aligned}
||u_y||_{L^\infty} & \leq \left(\int u_y^2 + u_{xxy}^2 + u_{yy}^2\right)^{1/2} \leq
||u||_{X^1}.\end{aligned}$$ Therefore, $$\partial_t \int \foned u_{xy}^2 + 3 \int (\foned)_x u_{xxy}^2 \leq \int (\foned)_x u_{yy}^2
+ C \int \foned u_{xx}^2 + C \int \foned u_{xy}^2.$$

\(c) [**The subcase $\alpha = (0,2)$.**]{} Applying $\partial_y^2$ to (2.1), multiplying by $u_{yy}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned}
& 2 \int u_{yy} u_{yyt} + 2 \int u_{yy}u_{xxxyy} - 2 \int u_{yy} \partial_x^{-1}u_{yyyy} \\
& \qquad + 2 \int u_{yy} u_{xyy} + 2 \int u_{yy}(uu_x)_{yy} = 0.\end{aligned}$$ Integrating by parts gets $$\partial_t \int u_{yy}^2 \leq 2 \left|\int u_{yy}(uu_x)_{yy}\right|.$$ Now $$\begin{aligned}
  \left|\int u_{yy} (uu_x)_{yy}\right| & = \left|\int u_{yy}(u_{yy}u_x + 2u_y u_{xy} + uu_{xyy})
  \right| \\
  & \leq ||u_x||_{L^\infty} \int u_{yy}^2 + ||u_y||_{L^\infty} \left(\int u_{yy}^2 + u_{xy}^2
  \right) \\
  & \leq ||u||_{X^0}\int u_{yy}^2 + C||u||_{X^1} \int (u_{yy}^2 + u_{xy}^2).\end{aligned}$$

Now combining these estimates from (a), (b) and (c) above, we have $$\begin{aligned}
  & \partial_t \int (\ftwod u_{xx}^2 + \foned u_{xy}^2 + u_{yy}^2) + 3 \int \left[(\ftwod)_x
  u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\
  & \qquad \leq C \int (\ftwod + \foned) u_{xx}^2
  + C \int ((\ftwod)_x + \foned + 1) u_{xy}^2 + \int ((\foned)_x + 1) u_{yy}^2,\end{aligned}$$ where $C$ depends only on the norm of $\phi \in X^1(\mathbb R^2)$. Since $\ftwod$ approximates $f_2 \in W_{0\;2\;0}$ and $\foned$ approximates $f_1
\in W_{0\;1\;0}$, we can choose $\ftwod, \foned$ such that $(\ftwod)_x \leq C \foned$, etc. Therefore, $$\begin{aligned}
  & \partial_t \int (\ftwod u_{xx}^2 + \foned u_{xy}^2 + u_{yy}^2) + 3 \int \left[(\ftwod)_x
  u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\
  & \qquad \leq C \int (\ftwod u_{xx}^2+ \foned u_{xy}^2 + u_{yy}^2).\end{aligned}$$ Integrating with respect to $t$, we have $$\begin{aligned}
  & \int (\ftwod(\cdot,t) u_{xx}^2 + \foned(\cdot,t) u_{xy}^2 + u_{yy}^2) +
  3 \int_0^t\int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \\
  & \leq \int (\ftwod(\cdot,0) \phi_{xx}^2 + \foned(\cdot,0) \phi_{xy}^2 + \phi_{yy}^2)
  + C \int_0^t \int (\ftwod u_{xx}^2+ \foned u_{xy}^2 + u_{yy}^2).\end{aligned}$$ Further, integrating by parts and using the fact that $f_1$ approximates $\foned \approx x$ for $x>1$, we note that $$\int \foned \phi_{xy}^2 \leq C \int \ftwod \phi_{xx}^2 + C \int \phi_{yy}^2.$$ Therefore, by Gronwall’s inequality $$\begin{aligned}
  & \sup_{0 \leq t \leq T} \int (\ftwod(\cdot,t) u_{xx}^2 + \foned(\cdot,t) u_{xy}^2 + u_{yy}^2) +
  3 \int_0^T\int \left[(\ftwod)_x u_{xxx}^2 + (\foned)_x u_{xxy}^2\right] \leq C\end{aligned}$$ where $C$ does not depend on $\delta$ but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2)
\cap \widetilde H_x^2(W_{0\;2\;0})$. Consequently, we can pass to the limit and conclude that $$\begin{aligned}
  & \sup_{0 \leq t \leq T} \int (f_2(\cdot,t) u_{xx}^2 + f_1(\cdot,t) u_{xy}^2 + u_{yy}^2) +
  3 \int_0^T\int \left[g_2 u_{xxx}^2 + g_1 u_{xxy}^2\right] \leq C.\end{aligned}$$

\(a) [**The subcase $\alpha = (3,0)$.**]{} We choose our weight functions such that $\fthreed$ approximates $f_3 \in W_{0\;3\;0}$. Applying $\partial_x^3$ to (2.1), multiplying by $\fthreed u_{xxx}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned}
& 2 \int \fthreed u_{xxx}u_{xxxt} + 2 \int \fthreed u_{xxx}u_{xxxxxx} - 2 \int \fthreed
u_{xxx}u_{xxyy} \\
& \qquad + 2 \int \fthreed u_{xxx}u_{xxxx} + 2 \int \fthreed u_{xxx}(uu_x)_{xxx} = 0.\end{aligned}$$ Integrating by parts gets $$\begin{aligned}
  & \partial_t \int \fthreed u_{xxx}^2 + 3 \int (\fthreed)_x u_{xxxx}^2 \\
  & \qquad = \int (\fthreed)_x u_{xxy}^2 + \int [\partial_t \fthreed + \partial_x^3
  \fthreed + \partial_x \fthreed]u_{xxx}^2 - 2 \int \fthreed u_{xxx}(uu_x)_{xxx} \\
  & \qquad \leq \int (\fthreed)_x u_{xxy}^2 + c \int \fthreed u_{xxx}^2 + 2 \left|
  \int \fthreed u_{xxx} (uu_x)_{xxx} \right|.\end{aligned}$$ Moreover $$\begin{aligned}
  \left|\int \fthreed u_{xxx}(uu_x)_{xxx} \right|
    & = \left|\int \fthreed [3u_{xx}^2 + 4u_x u_{xxx} + uu_{xxxx}] u_{xxx} \right| \\
  & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx} \right| + 4 \left|\int \fthreed u_x u_{xxx}^2
  \right| + \left|\int \fthreed u u_{xxx} u_{xxxx} \right| \\
  & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx}\right| + c ||u_x||_{L^\infty}\int \fthreed
  u_{xxx}^2 + c \left|\int (\fthreed u)_x u_{xxx}^2 \right| \\
  & \leq 3 \left|\int \fthreed u_{xx}^2 u_{xxx} \right| + c ||u||_{X^0} \int \fthreed u_{xxx}^2 \\
  & \qquad + c (||u||_{L^\infty(\mathbb R^2)} + ||u_x||_{L^\infty(\mathbb R^2)})\int
  \fthreed u_{xxx}^2 \\
  & \leq C \left|\int \fthreed u_{xx}^2 u_{xxx}\right| + c||u||_{X^0(\mathbb R^2)}\int \fthreed
  u_{xxx}^2.\end{aligned}$$ Now we estimate the first term on the right-hand side.

Let $A_1 = \left\{x \in \mathbb R: x > 1\right\} \times \mathbb R
\subseteq \mathbb R^2$. $$\begin{aligned}
  \left|\int_{A_1} \fthreed u_{xx}^2 u_{xxx} \right|
    & = C \left|\int_{A_1} (\fthreed)_x u_{xx}^3 \right| \\
    & \leq C ||u_{xx}||_{L^\infty(A_1)} \int_{A_1} (\fthreed)_x u_{xx}^2 \\
    & \leq C \left(\int_{A_1} u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2\right)^{1/2}
    \int_{A_1} \ftwod u_{xx}^2 \\
    & \leq \epsilon \int_{A_1} (u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2) + C\left(\int_{A_1}
    \ftwod u_{xx}^2\right)^2.\end{aligned}$$ Now the terms involving $u_{xx}$ have been bounded by the previous step in the induction. Therefore, we conclude that $$\left|\int_{A_1} \fthreed u_{xx}^2 u_{xxx} \right| \leq C + \epsilon \int_{A_1} (u_{xxxx}^2
+ u_{xxy}^2).$$

$x < -1$. Let $A_{-1} = \left\{x \in \mathbb R: x < -1\right\} \times
\mathbb R \subseteq \mathbb R^2$. We use the fact that $\fthreed \approx c$ to show $$\begin{aligned}
  \left|\int_{A_{-1}} \fthreed u_{xx}^2 u_{xxx} \right|
  & \leq c \left|\int_{A_{-1}} u_{xx}^2 u_{xxx} \right| \\
  & \leq c \left(\int_{A_{-1}} u_{xx}^4\right)^{1/2}\left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\
  & \leq C \left(\int_{A_{-1}}[u_{xx}^2 + u_{xxx}^2
  + u_{xy}^2]\right)\left(\int_{A_{-1}} u_{xxx}^2 \right)^{1/2} \\
  & \leq c ||u||_{X^0}^3.\end{aligned}$$ Combining these estimates for the subcase $\alpha = (3,0)$, yields $$\begin{aligned}
  & \int \fthreed(\cdot,t) u_{xxx}^2 + 3 \int_0^t \int (\fthreed)_x u_{xxxx}^2 \\
  & \leq
  \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c
  \int_0^t \int \fthreed u_{xxx}^2 + \epsilon \int_0^t \int_{A_1} u_{xxxx}^2 \\
  & \leq
  \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c
  \int_0^t \int \fthreed u_{xxx}^2 + \epsilon \int_0^t \int (\fthreed)_x u_{xxxx}^2\end{aligned}$$ Therefore, $$\begin{aligned}
  & \int \fthreed(\cdot,t) u_{xxx}^2 + 3 \int_0^t \int (\fthreed)_x u_{xxxx}^2 \\
    & \leq  \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + \int_0^t\int (\fthreed)_x u_{xxy}^2 + c
  \int_0^t \int \fthreed u_{xxx}^2 \\
  & \leq \int \fthreed(\cdot,0) \phi_{xxx}^2 + C + C \int_0^t \int \ftwod u_{xxy}^2 +
  C \int_0^t \int \fthreed u_{xxx}^2.\end{aligned}$$

\(b) [**The subcase $\alpha = (2,1)$.**]{} In this case, we take $\ftwod$ approximating $f_2 \in W_{0\;2\;0}$. Apply $\partial_x^2 \partial_y$ to (2.1), multiply by $\ftwod u_{xxy}$ and integrating over $\mathbb R^2$, we have $$\begin{aligned}
  & \partial_t \int \ftwod u_{xxy}^2 + 3 \int (\ftwod)_x u_{xxxy}^2 \\
  & \leq \int (\ftwod)_x u_{xyy}^2 + \int \ftwod u_{xxy}^2 + 2\left|\int
  \ftwod u_{xxy} (uu_x)_{xyy}\right|.\end{aligned}$$ Now $$\begin{aligned}
  \left|\int \ftwod u_{xxy}(uu_x)_{xxy} \right| & = \left|\int \ftwod
  u_{xxy} [3u_{xy}u_{xx} + 3u_x u_{xxy} + u_y u_{xxx} + uu_{xxxy}]\right| \\
  & \leq C \int \ftwod u_{xxy}u_{xy} u_{xx} + \int \ftwod u_{xxy}u_y u_{xxx}
  + C||u||_{X^0} \int \ftwod u_{xxy}^2.\end{aligned}$$ The second term on the right-hand side satisfies $$\begin{aligned}
\int \ftwod u_{xxy} u_y u_{xxx} & \leq C ||u_y||_{L^\infty} \int \ftwod (u_{xxx}^2 + u_{xxy}^2) \\
& \leq C ||u||_{X^1} \int (\fthreed u_{xxx}^2 + \ftwod u_{xxy}^2).\end{aligned}$$ For the first term on the right-hand side, we consider two cases. First, for $A_1$, $$\begin{aligned}
  \int_{A_1} \ftwod u_{xxy}u_{xy} u_{xx} & \leq ||u_{xy}||_{L^\infty(A_1)} \left(\int_{A_1} \ftwod
  u_{xxy}^2\right)^{1/2} \left(\int_{A_1} \ftwod u_{xx}^2\right)^{1/2} \\
  & \leq \epsilon \left(\int_{A_1} u_{xy}^2 + u_{xxxy}^2 + u_{xyy}^2\right) + C
   \int_{A_1} \ftwod u_{xxy}^2 \\
   & \leq C + \epsilon \int (\ftwod)_x u_{xxxy}^2 + C \int \foned u_{xyy}^2 + C
   \int \ftwod u_{xxy}^2.\end{aligned}$$

Then for $A_{-1}$, $$\begin{aligned}
  \int_{A_{-1}} \ftwod u_{xxy} u_{xy} u_{xx} & = \int_{A_{-1}} u_{xxy} u_{xy} u_{xx} \\
  & = C \int_{A_{-1}} u_{xy}^2 u_{xxx} \\
  & \leq C \left(\int_{A_{-1}} u_{xy}^4\right)^{1/2} \left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\
  & \leq C \left(\int_{A_{-1}} u_{xy}^2 + u_{xxy}^2 + u_{yy}^2\right)\left(\int_{A_{-1}} u_{xxx}^2
  \right)^{1/2} \\
  & \leq C(||u||_{X^0}) \left(1 + \int \ftwod u_{xxy}^2\right)\end{aligned}$$

Combining these estimates and integrating with respect to $t$, we have $$\begin{aligned}
\int \ftwod(\cdot,t) u_{xxy}^2 + 3 \int_0^t \int (\ftwod)_x u_{xxxy}^2
& \leq C + \int \ftwod(\cdot,0) \phi_{xxy}^2 +
\epsilon \int_0^t \int (\ftwod)_x u_{xxxy}^2 \\
& \qquad + \int_0^t \int \ftwod u_{xxy}^2
+ C \int_0^t \int \foned u_{xyy}^2.\end{aligned}$$ Therefore, $$\int \ftwod(\cdot,t) u_{xxy}^2 + 3 \int_0^t \int (\ftwod)_x u_{xxxy}^2
 \leq C + \int \ftwod(\cdot,0) \phi_{xxy}^2  + \int_0^t \int \ftwod u_{xxy}^2
+ C \int_0^t \int \foned u_{xyy}^2.$$

\(c) [**The subcase $\alpha = (1,2)$.**]{} We take our weight function $\foned$ approximating $f_1 \in W_{0\;1\;0}$. Applying $\partial_x\partial_y^2$ to (2.1), multiplying by $\foned u_{xyy}$ and integrating over $\mathbb R^2$, we get $$\begin{aligned}
  & \partial_t \int \foned u_{xyy}^2 + 3 \int (\foned)_x u_{xxyy}^2 \\
  & \leq c \int (\foned)_x u_{yyy}^2 + \int \foned u_{xyy}^2 + 2 \left|\int \foned
  u_{xyy} (uu_x)_{xyy}\right|.\end{aligned}$$

Now $$\begin{aligned}
  2 \left|\int \foned u_{xyy} (uu_x)_{xyy} \right| & = 2 \left|\int \foned u_{xyy}
  (2u_{xy}^2 + 2u_x u_{xyy} + u_{yy}u_{xx} + 2u_y u_{xxy} + uu_{xxyy})\right|.\end{aligned}$$ The first term on the right-hand side satisfies $$\int \foned u_{xyy}u_{xy}^2 = C \int \foned (u_{xy}^3)_y = 0,$$ since $(\foned)_y = 0$. Integrating by parts, it is clear that the second and fifth terms on the right-hand side are bounded by $$C ||u||_{X^0}\int \foned u_{xyy}^2.$$ The fourth term on the right-hand side is bounded by $$C||u_y||_{L^\infty} \left(\int \foned u_{xyy}^2 + \int \foned u_{xxy}^2\right).$$ For the third-term on the right-hand side, we consider the cases when $x > 1$ and $x < -1$ separately. First, for $x > 1$, we have $$\begin{aligned}
  \int_{A_1} \foned u_{xyy} u_{yy} u_{xx} & \leq ||u_{yy}||_{L^\infty(A_1)} \left(\int_{A_1}
  \foned u_{xyy}^2\right)^{1/2} \left(\int_{A_1} \foned u_{xx}^2\right)^{1/2} \\
  & \leq C \left(\int_{A_1} u_{yy}^2 + u_{xxyy}^2 + u_{yyy}^2\right)^{1/2} \left(\int_{A_1}
  \foned u_{xyy}^2\right)^{1/2} \\
  & \leq C + \epsilon \int_{A_1} u_{xxyy}^2 + C \int_{A_1} \foned u_{xyy}^2 + C \int_{A_1} u_{yyy}^2\end{aligned}$$ where we have used the fact that $\int \foned u_{xx}^2$ was bounded on the previous step of the induction. We will bound the $\epsilon$ term back on the left-hand side. For $x < -1$, we have $$\begin{aligned}
  \int_{A_{-1}} \foned u_{xyy}u_{yy}u_{xx} & \approx \int \chi_{[x < -1]}u_{yy}^2 u_{xxx} \\
  & \leq \left(\int_{A_{-1}} u_{yy}^4\right)^{1/2} \left(\int_{A_{-1}} u_{xxx}^2\right)^{1/2} \\
  & \leq C\int_{A_{-1}} u_{yy}^2 + u_{xyy}^2 + (\partial_x^{-1} u_{yy})^2 \\
  & \leq C\end{aligned}$$ where $C$ depends only on the norm of $u \in X^1(\mathbb R^2)$. Combining these estimates and integrating with respect to $t$, we have $$\begin{aligned}
  \int \foned(\cdot,t) u_{xyy}^2 + 3 \int_0^t \int (\foned)_x u_{xxyy}^2 \leq
  \int \foned(\cdot,0) \phi_{xyy}^2 + C \int_0^t \int \foned u_{xyy}^2 + C \int_0^t \int u_{yyy}^2.\end{aligned}$$

\(d) [**The subcase $\alpha = (0,3)$.**]{} In this case we apply $\partial_y^3$ to (2.1), multiply by $u_{yyy}$ and integrate over $\mathbb R^2$. We have $$\partial_t \int u_{yyy}^2 \leq \int u_{yyy}^2 + 2 \left|\int u_{yyy}(uu_x)_{yyy}\right|.$$ Now $$\left|\int u_{yyy}(uu_x)_{yyy} \right| = \left|\int u_{yyy}(u_xu_{yyy} + 3u_{yy}u_{xy} +
3u_y u_{xyy} + uu_{xyyy})\right|$$ Integrating by parts as necessary, we see that the first and fourth terms on the right-hand side are bounded by $$||u||_{X^0} \int u_{yyy}^2.$$ The second term on the right-hand side is bounded by $$\begin{aligned}
& \left(\int u_{yy}^4\right)^{1/4} \left(\int u_{xy}^4\right)^{1/4} \left(\int u_{yyy}^2
\right)^{1/2} \\
& \leq \left(\int u_{yy}^2 + u_{xyy}^2 + (\partial_x^{-1}u_{yyy})^2\right)^{1/2}
\left(\int u_{xy}^2 + u_{xxy}^2 + u_{yy}^2\right)^{1/2} \left(\int u_{yyy}^2\right)^{1/2} \\
& \leq C||u||_{X^1(\mathbb R^2)}\left(\int u_{yyy}^2 \right)^{1/2} \\
& \leq C + C \int u_{yyy}^2.\end{aligned}$$ The third term on the right-hand side is bounded by $$\begin{aligned}
& C||u_y||_{L^\infty} \left(\int u_{xyy}^2\right)^{1/2} \left(\int u_{yyy}^2\right)^{1/2} \\
& \leq C \left(\int u_y^2 + u_{xxy}^2 + u_{yy}^2 \right)^{1/2} \left(\int u_{xyy}^2\right)^{1/2}
\left(\int u_{yyy}^2\right)^{1/2} \\
& \leq C ||u||_{X^1}^2 \left(\int u_{yyy}^2\right)^{1/2} \\
& \leq C + C \int u_{yyy}^2.\end{aligned}$$

Combining the estimates above and integrating with respect to $t$, we have $$\begin{aligned}
  \int u_{yyy}^2 \leq \int \phi_{yyy}^2 + C + C \int_0^t \int u_{yyy}^2\end{aligned}$$ where $C$ depends only on the norm of $\phi$ in $X^1(\mathbb R^2)$.

Combining the estimates above for (a), (b), (c) and (d) and applying Gronwall’s inequality, we conclude that $$\begin{aligned}
  & \sup_{0 \leq t \leq T}\int (\fthreed u_{xxx}^2 + \ftwod u_{xxy}^2 + \foned u_{xyy}^2 + u_{yyy}^2)
  \\
  & + 3 \int_0^T \int ((\fthreed)_x u_{xxxx}^2 + (\ftwod)_x u_{xxxy}^2 + (\foned)_x u_{xxyy}^2)
  \leq C\end{aligned}$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2)
\cap \widetilde H_x^3(W_{0\;3\;0})$. Consequently, we can pass to the limit and conclude that $$\sup_{0 \leq t \leq T} \int (f_3 u_{xxx}^2 + f_2 u_{xxy}^2 + f_1 u_{xyy}^2 + u_{yyy}^2) +
  3 \int_0^T \int (g_3 u_{xxxx}^2 + g_2 u_{xxxy}^2 + g_1 u_{xxyy}^2)
  \leq C.$$

In this case, we take $f_{\alpha_1,\delta}$ approximating $\faone \in W_{0\;\alpha_1\;0}$. We apply $\pa$ to (2.1), multiply by $f_{\alpha_1,\delta}\pa u$ and integrate over $\mathbb R^2$. We need to get a bound on $$\left|\int_0^T \int f_{\alpha_1,\delta} (\pa u) \pa (uu_x)\right|.$$ In Lemma 6.2 below, we prove that $$\left|\int_0^t \int \fad (\pa u)\pa(uu_x)\right| \leq C + C \sum_{\gamma_1 + \gamma_2 =
  j} \int_0^t \int \fgd (\partial^\gamma u)^2$$ where $C$ depends only on terms bounded in the previous step of the induction. Consequently, we have that $$\sup_{0\leq t\leq T} \sum_{|\alpha|=j} f_{\alpha_1,\delta} (\pa u)^2 + \sum_{|\alpha|=j} \int_0^T \int
  (f_{\alpha_1,\delta})_x (\pa u_x)^2 \leq C,$$ where $C$ does not depend on $\delta$, but only on $T$ and the norm of $\phi \in X^1(\mathbb R^2)
\cap \widetilde H_x^j(W_{0\;j\;0})$. Passing to the limit, we get the desired estimate, namely, $$\sup_{0\leq t\leq T} \sum_{|\alpha|=j} \faone (\pa u)^2 + \sum_{|\alpha|=j} \int_0^T \int
  \gaone (\pa u_x)^2 \leq C.$$ $\square$

In order to get bounds on the left-hand side of , we use the fact that every term in the integrand is of the form $$\fad (\pa u)(\partial^r u)(\partial^s u_x)$$ where $r_i + s_i = \alpha_i$. Before showing the bounds on each of the terms in the integrand we point one bound we will be using frequently: $$||\partial^\gamma u||_{L^\infty} \leq C \left(\int (\partial^\gamma u)^2 + (\partial^\gamma u_{xx})^2
  + (\partial^\gamma u_y)^2\right)^{1/2}.$$

In this case, $|r| = 0$ and $s = \alpha$. Therefore, $$\begin{aligned}
  \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned u (\pa u)(\pa u_x) \\
  & = - \frac 12 \int [\faoned u]_x (\pa u)^2 \\
  & \leq C ||u||_{X^0} \int \faoned (\pa u)^2.\end{aligned}$$

Therefore, $|r| = 1$. We have two subcases below:

\(a) [**The subcase $r = (1,0)$.**]{} In this case, $$\begin{aligned}
  \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_x (\pa u) \\
  & \leq C||u_x||_{L^\infty} \int \faoned (\pa u)^2.\end{aligned}$$

\(b) [**The subcase $r = (0,1)$.**]{} In this case, $$\begin{aligned}
  \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_y
  (\partial_x^{\alpha+1} \partial_y^{\beta-2} u) \\
  & \leq C||u_y||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \faoned (\partial_x
  ^{\alpha_1+1}\partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\
  & \leq C||u||_{X^1} \left[\int \faoned (\pa u)^2 + \int f_{\alpha+1,\delta}(\partial_x^{\alpha_1+1,
  \alpha_2-1}u)^2\right].\end{aligned}$$

We consider three subcases below:

\(a) [**The subcase $r = (2,0)$.**]{} In this case $$\begin{aligned}
  \int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & = \int \faoned (\pa u)u_{xx}
  (\partial_x^{\alpha_1-1}
  \partial_y^{\alpha_2} u) \\
  & \leq C||\foned u_{xx}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2}
  \left(\int f_{\alpha_1-1,\delta}(\partial_x^{\alpha_1-1}\partial_y^{\alpha_2}u)^2\right)^{1/2}.\end{aligned}$$ Now $$\foned u_{xx} \approx \left\{
\begin{array}{ll}
x u_{xx} & \quad x > 1 \\
u_{xx} & \quad x < -1.
\end{array}
\right.$$ For $x < -1$, we use the fact that $$\begin{aligned}
||u_{xx}||_{L^\infty(A_{-1})} & \leq C \left(\int_{A_{-1}} (u_{xx}^2 + u_{xxxx}^2 + u_{xxy}^2)
\right)^{1/2} \\
& \leq C ||u||_{X^1}.\end{aligned}$$

For $x > 1$, we use the fact that $$\begin{aligned}
\foned u_{xx} & = (\foned u)_{xx} - (\foned)_{xx} u - 2(\foned)_x u_x \\
& = (\foned u)_{xx} - (\foned)_{xx} u - 2 ((\foned)_xu)_x + 2 (\foned)_{xx} u.\end{aligned}$$ Therefore, $$\begin{aligned}
||\foned u_{xx} ||_{L^\infty} & \leq ||(\foned u)_{xx}||_{L^\infty} + C ||((\foned)_x u)_x||_{L^\infty}
+ C||(\foned)_{xx}u||_{L^\infty} \\
& \leq C \left(\int ((\foned u)_{xx})^2 + ((\foned u)_{xxxx})^2 + ((\foned u)_{xxy})^2\right)^{1/2} \\
& + C \left(\int (((\foned)_xu)_x)^2 + (((\foned)_x u)_{xxx})^2 + (((\foned)_x u)_{xy})^2
\right)^{1/2} \\
& + C\left(\int ((\foned)_{xx}u)^2 + (((\foned)_{xx}u)_{xx})^2 + (((\foned)_{xx}u)_y)^2
\right)^{1/2} \\
& \leq C + C\int u^2 + u_x^2 + \foned u_{xx}^2 + u_{xxx}^2 + \foned u_{xxxx}^2 + u_y^2 + u_{xy}^2
+ \foned u_{xxy}^2 \\
& \leq C + C \sum_{|\gamma| \leq j} \int \fgd (\partial^\gamma u)^2.\end{aligned}$$

\(b) [**The subcase $r=(1,1)$.**]{} Then $$\begin{aligned}
\int \faoned (\pa u)(\partial^r u)(\partial^su_x) & = \int \faoned (\pa u)u_{xy} (\partial_x^{\alpha_1}
\partial_y^{\alpha_2-1} u) \\
& \leq ||u_{xy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \faoned
(\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\
& \leq C||u||_{X^1} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int \fad
(\partial_x^{\alpha_1} \partial_y^{\alpha_2-1}u)^2\right)^{1/2} \\
& \leq C + C \sum_{|\gamma| \leq j} \int \fgd (\partial^\gamma u)^2\end{aligned}$$ where $C$ depends only on the bounds in the statement of the theorem.

\(c) [**The subcase $r=(0,2)$.**]{} Then $$\begin{aligned}
  \int \faoned (\pa u)(\partial^r u)(\partial^su_x) & = \int \faoned (\pa u)u_{yy}
  (\partial_x^{\alpha_1+1}\partial_y^{\alpha_2-2}u) \\
  & \leq ||u_{yy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \left(
  \int \faoned (\partial_x^{\alpha_1+1} \partial_y^{\alpha_2-2} u)^2\right)^{1/2} \\
  & \leq C ||u_{yy}||_{L^\infty} \left(\int \faoned (\pa u)^2\right)^{1/2} \\
  & \leq C \left(\int u_{yy}^2 + u_{xxyy}^2 + u_{xyy}^2\right)^{1/2} \left(\int \faoned
  (\pa u)^2\right)^{1/2} \\
  & \leq C + C\sum_{|\gamma| = j}\int \fgd (\partial^\gamma u)^2\end{aligned}$$ where $C$ depends only on the bounds in the statement of the theorem.

In this case we consider $x > 1$ and $x < -1$ separately. First, for $x < -1$, we have $$\begin{aligned}
  \int_{A_{-1}} \faoned (\pa u)(\partial^r u)(\partial^s u_x)
  & \leq \int \left(\int_{A_{-1}} (\pa u)^2
  \right)^{1/2} \left(\int_{A_{-1}} (\partial^r u)^4\right)^{1/4}
  \left(\int_{A_{-1}} (\partial^s u_x)^4\right)^{1/4}.\end{aligned}$$ Now $$\left(\int_{A_{-1}} (\partial^r u)^4\right)^{1/4} \leq \left(\int_{A_{-1}} (\partial^r u)^2
+ (\partial^r u_x)^2 + (\partial^r u_y)^2\right)^{1/2}$$ and $|r| = 3$ implies each of these terms is bounded by $C$ where $C$ depends only on $$\int \fgd (\partial^\gamma u)^2 \qquad \text{for } |\gamma| \leq j-1$$ since $j \geq 5$ and each of these terms has derivatives of order $\leq 4$. Also, $$\left(\int_{A_{-1}} (\partial^s u_x)^4 \right)^{1/4} \leq \left(\int_{A_{-1}}
(\partial^su_x)^2 + (\partial^su_{xx})^2 + (\partial^s u_{xy})^2 \right)^{1/2}$$ and $|s| = j-3$. Therefore, each of these terms has order at most $j-1$ and thus bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$ Therefore, for $x < -1$, we have $$\int_{A_{-1}} \faoned (\pa u)(\partial^r u)(\partial^s u_x)
\leq C + C \int_{A_{-1}} \faoned (\pa u)^2,$$ where $C$ depends only on the terms in the statement of the theorem.

Now for $x > 1$, we have $$\begin{aligned}
\int_{A_1} \faoned (\pa u)(\partial^r u)(\partial^s u_x)
& \leq \left(\int_{A_1} \faoned (\pa u)^2
\right)^{1/2} \left(\int_{A_1} (f_{r_1/2,\delta} \partial^ru)^4\right)^{1/4} \\
& \times \left(\int_{A_1} (f_{(s_1+1)/2,\delta}
\partial^su_x)^4\right)^{1/4}.\end{aligned}$$ since $r_1 + s_1 = \alpha_1$. Now $$\begin{aligned}
\left(\int_{A_1} (f_{r_1/2,\delta} \partial^r u)^4 \right)^{1/4} & \leq
\left(\int_{A_1} (f_{r_1/2,\delta} \partial^r u)^2 + ((f_{r_1/2,\delta} \partial^ru)_x)^2
+ ((f_{r_1/2,\delta} \partial^r u)_y)^2\right)^{1/2} \\
& \leq C \left(\int_{A_1} f_{r_1,\delta} (\partial^ru)^2 + f_{r_1,\delta} (\partial^r u_x)^2
+ f_{r_1,\delta}(\partial^ru_y)^2\right)^{1/2}.\end{aligned}$$ Since $|r| = 3$, each of the terms above has order at most four, and, therefore, is bounded by a constant $C$ which depends only on $$\int \fgd (\partial^\gamma u)^2 \qquad \text{for } |\gamma| \leq 4 \leq j-1,$$ since here we are assuming $j \geq 5$.

Similarly, $$\begin{aligned}
  & \left(\int (f_{(s_1+1)/2,\delta}(\partial^s u_x))^4\right)^{1/4} \\
  & \leq C
  \left(\int (f_{(s_1+1)/2,\delta}(\partial^s u_x))^2+((f_{(s_1+1)/2,\delta}(\partial^su_x))_x)^2
  + ((f_{(s_1+1)/2,\delta}(\partial^su_x))_y)^2\right)^{1/2} \\
  & \leq C \left(\int f_{s_1+1,\delta}[(\partial^su_x)^2 + (\partial^su_{xx})^2+ (\partial^s
  u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Since $|s| = j-3$ each of these terms is of order at most $j-1$. Therefore, each of these terms is bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$

In this case, $|s| = 1$. Therefore, either $s = (1,0)$ or $s = (0,1)$. For $s = (1,0)$, we have $$\begin{aligned}
\int \faoned (\pa u)(\partial^r u) (\partial^s u_x) & = \int \faoned (\pa u)(\partial^r u)
u_{xx} \\
& \leq \left(\int \faoned (\pa u)^2\right)^{1/2} \left(\int (f_{r_1/2,\delta} \partial^r u)^4
\right)^{1/4} \left(\int (\foned u_{xx})^4\right)^{1/4}.\end{aligned}$$

Now $$\begin{aligned}
  \left(\int (f_{r_1/2,\delta}(\partial^r u))^4\right)^{1/4} & \leq \left(\int f_{r_1,\delta}
  [(\partial^r u)^2+ (\partial^r u_x)^2 + (\partial^r u_y)^2]\right)^{1/2}.\end{aligned}$$ We note that $|r| = 3$. Therefore, each of these terms is at most of order $4 = j$. Further, $$\begin{aligned}
  \left(\int (\foned u_{xx})^4\right)^{1/4} & \leq C \left(\int \ftwod[u_{xx}^2 + u_{xxx}^2
  + u_{xxy}^2]\right)^{1/2}.\end{aligned}$$ We note that each of these terms is of order at most $j-1$. Combining these estimates, we conclude that $$\int \faoned (\pa u)(\partial^r u) (\partial^s u_x) \leq C + C \sum_{|\gamma| = j}
\int \fgd (\partial^\gamma u)^2,$$ where $C$ depends only on $$\sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$

The case in which $s = (0,1)$ is handled similarly.

In this case, we bound the terms as follows: $$\begin{aligned}
\int \faoned (\pa u)(\partial^r u)(\partial^s u_x) & \leq ||f_{(s_1+1)/2,\delta}\partial^su_x
||_{L^\infty} \left(\int \faoned(\pa u)^2\right)^{1/2} \left(\int f_{r_1,\delta} (\partial^r u)^2
\right)^{1/2}.\end{aligned}$$ Now $$\begin{aligned}
  ||f_{(s_1+1)/2,\delta}(\partial^su_x)||_{L^\infty} & \leq C\left(\int f_{s_1+1,\delta}[
  (\partial^s u_x)^2 + (\partial^s u_{xxx})^2 + (\partial^s u_{xy})^2]\right)^{1/2}.\end{aligned}$$ Since $|s| \leq j-4$, all of these terms are of order at most $j-1$. Therefore, each of these terms is bounded by $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2$$ and, therefore, $$\int \faoned (\pa u) (\partial^r u)(\partial^s u_x) \leq C + C \sum_{|\gamma| = j}
\int \fgd (\partial^\gamma u)^2.$$ where $C$ depends only on $$C \sum_{|\gamma| \leq j-1} \int \fgd (\partial^\gamma u)^2.$$$\square$

Main Theorem
============

In this section we state and prove our main theorem, which states that if the initial data $\phi$ possesses certain regularity and sufficient decay at infinity, then the solution $u(t)$ will be smoother than $\phi$. In particular if the initial data satisfies $$\int \phi^2 + (1 + x_+^L) (\partial_x^L \phi)^2 + (\partial_y^L \phi)^2 < \infty,$$ then the solution will [*gain*]{} $L$ derivatives in $x$. More specifically, $$\int_0^T \int t^{L-1}(1+e^{\sigma x_-}) (\partial_x^{2L} u)^2 < \infty$$ for $\sigma > 0$ arbitrary, where $T$ is the existence time of the solution.\
(Main Theorem). [*Let $T>0$ and let $u$ be the solution of in the region $[0,\,T]\times\mathbb{R}^{2}$ such that $u\in
L^{\infty}([0,\,T]:\,{\cal Z}_{L})$ for some $L\geq 2.$ Then*]{} $$\begin{aligned}
\label{e701}\sup_{0\leq t\leq
T}\int_{\mathbb{R}^{2}}f_\alpha\,(\pa u)^{2}\,dx\,dy +
\int_{0}^{T}\int_{\mathbb{R}^{2}}g_\alpha\,(\pa u_x)^{2}\,dx\,dy\,dt \leq C\end{aligned}$$ [*for $L+1\leq |\alpha| \leq 2L - 1$, $2L-|\alpha|-\alpha_2 \geq 1$ where $f_\alpha \in
W_{\sigma,\;2L - |\alpha|-\alpha_2,\;|\alpha|-L}$ and $g_\alpha\in
W_{\sigma,\;2L - |\alpha| - \alpha_2 - 1,\;|\alpha|-L},$ $\sigma>0$ arbitrary.*]{}\
\
[*Proof.*]{} By assumption, $u\in L^{\infty}([0,\,T]:\,{\cal
Z}_{L}).$ Therefore $u_{t}\in
L^{\infty}([0,\,T]:\,L^{2}(\mathbb{R}^{2})),$ then $u\in
C([0,\,T]:\,L^{2}(\mathbb{R}^{2}))\cap C_{w}([0,\,T]:\,{\cal
Z}_{L}).$ Hence $u:[0,\,T]\rightarrow {\cal Z}_{L}$ is a weakly continuous function. In particular, $u(\,\cdot\,,\,\cdot\,,t)\in
{\cal Z}_{L}$ for every $t.$ Let $t_{0}\in (0,\,T)$ and $u(\,\cdot\,,\,\cdot\,,t_{0})\in {\cal Z}_{L},$ then there are $\{\phi^{(n)}\}\subseteq C_{0}^{\infty}(\mathbb{R}^{2})$ such that $\partial_{x}^{-1}\phi_{yy}^{(n)}$ are in $C_{0}^{\infty}(\mathbb{R}^{2})$ and $\phi^{(n)}(\,\cdot\,,\,\cdot\,)\rightarrow
u(\,\cdot\,,\,\cdot\,,t_{0})$ in ${\cal Z}_L.$ Let $u^{(n)}$ be the unique solution of with initial data $\phi^{(n)}(x,\,y)$ at time $t=t_{0}.$ By Corollary 4.4, the solution $u^{(n)}\in L^{\infty}([t_{0},\,t_{0} +
\delta]:\,X^{1}(\mathbb{R}^{2}))$ for a time interval $\delta$ which not depend on $n.$ By Theorem 6.1, $u^{(n)}\in
L^{\infty}([t_{0},\,t_{0} + \delta]:\,{\cal Z}_L)$ and $$\begin{aligned}
\label{e702}\int_{t_{0}}^{t_{0} +
\delta}\int_{\mathbb{R}^{2}}g_{\alpha_1}\,(\pa
u_x)^{2}\,dx\,dy\,dt\leq C\end{aligned}$$ for $|\alpha|=L$, $\alpha_1 \neq 0$, where $g_{\alpha_1}\in W_{\sigma,\;\alpha_1 - 1,\;0}$ and $C$ depends only on the norm of $\phi^{(n)}\in {\cal Z}_{L}.$ Also by Theorem 6.1, we have (non-uniform) bounds on $$\begin{aligned}
\label{e703}\sup_{[t_{0},\,t_{0} + \delta]}\sup_{(x,y)}\left[(1 +
x^{+})^{k}|\partial_x^{\alpha_1} u^{(n)}(x,\,y,\,t)| + |\partial_y^{\alpha_2}
u^{(n)}(x,\,y,\,t)|\right] <+\infty\end{aligned}$$ for each $n,\,k$ and $\alpha_1,\,\alpha_2.$ Therefore, the a priori estimates in Lemma 5.1, are justified for each $u^{(n)}$ in the interval $[t_{0},\,t_{0} + \delta].$\
\
We start our induction with $|\alpha|=L+1$. In this case, we take $g_\alpha
\in W_{\sigma,\;L - 2 - \alpha_2,\;1}$ and let $f_\alpha =\frac{1}{3}\int_{-\infty}^{x}
g_\alpha(z,\,t)\,dz.$ We note that $2L-|\alpha|-\alpha_2 \geq 1$ by assumption. Therefore, $L-2-\alpha_2 \geq 0$. As shown in Lemma 5.1, we have the following bounds on the higher derivatives of $u^{(n)},$ $$\begin{aligned}
\label{e704}\sup_{[t_{0},\,t_{0} +
\delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u^{(n)})^{2}\,dx\,dy
+ \int_{t_{0}}^{t_{0} +
\delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u^{(n)}_x)^{2}\,dx\,dy\,dt\leq
C\end{aligned}$$ where $C$ depends only on the norm of $u^{(n)}\in
L^{\infty}([0,\,T]:\,{\cal Z}_{L})$ and the term in . We conclude, therefore, that the constant $C$ in depends only on $||\phi^{(n)}||_{{\cal Z}_{L}}.$ We continue this procedure inductively. For the $|\alpha|^{th}$ step, let $g_\alpha \in W_{\sigma,\;2L - |\alpha|
-\alpha_2 - 1,\;|\alpha|-L}$ for $\alpha_2 \leq 2L-|\alpha|-1$ and define $f_\alpha =\frac{1}{3}\int_{-\infty}^{x}g_\alpha(z,\,t)\,dz.$ The non-uniform bounds on $u^{(n)}$ in allows us to use Lemma 5.1 and our inductive hypothesis to conclude that $$\begin{aligned}
\sup_{[t_{0},\,t_{0} +
\delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u^{(n)})^{2}\,dx\,dy + \int_{t_{0}}^{t_{0} +
\delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u^{(n)}_x)^{2}\,dx\,dy\,dt\leq C\end{aligned}$$ where again $C$ does not depend on $n,$ but only on the norm of $\phi^{(n)}\in {\cal Z}_{L}.$ By Corollary 4.5, $$\begin{aligned}
u^{(n)}\stackrel{*}\rightharpoonup u\quad \mbox{weakly in}\quad
L^{\infty}([t_{0},\,t_{0} + \delta]:\,X^{1}(\mathbb{R}^{2})).\end{aligned}$$ Therefore, we can pass to the limit and conclude that $$\begin{aligned}
\label{e706}\sup_{[t_{0},\,t_{0} +
\delta]}\int_{\mathbb{R}^{2}}f_\alpha \,(\pa u)^{2}\,dx\,dy + \int_{t_{0}}^{t_{0} +
\delta}\int_{\mathbb{R}^{2}}g_\alpha \,(\pa u)^{2}\,dx\,dy\,dt\leq C.\end{aligned}$$ This proof is continued inductively up to $|\alpha| =2L - 1.$ Since $\delta$ does not depend on $n,$ this result is valid over the whole interval $[0,\,T].$ $\square$

[00]{}

M.J. Ablowitz & P.A. Clarkson, [*Solitons, Nonlinear Evolution Equations and Inverse Scattering*]{}, Cambridge University Press, Cambridge, 1991.

O. V. Besov, V. P. IL’IN and S. M. Nikolskii, [*Integral representations of functions and imbedding theorems,*]{} Vol. I, J. Wiley 1978.

J.C. Ceballos, M. Sepulveda and O. Vera, [*Gain in regularity for a coupled nonlinear Schrodinger system*]{}, Boletim da Sociedade Paranaense de Matematica. To appear.

A. Cohen, [*Solutions of the Korteweg-deVries equation from irregular dat*]{}, Duke Math. J. [**45**]{} (1978), 149-181.

P. Constantin and J.C. Saut, [*Local smoothing properties of dispersive equations*]{}, J. Amer. Math. Soc. [**1**]{} (1988), 413-439.

W. Craig & J. Goodman, [*Linear dispersive equations of airy type*]{}, J. Diff. Eqns. [**7**]{} (1990), 38-61.

W. Craig, T. Kappeler, & W. Strauss, [*Gain of regularity for equations of KdV type*]{}, Ann. Inst. H. Poincare [**9**]{} (1992), 147-186.

D. David, D. Levi & P. Winternitz, [*Solitons in shallow seas of variable depth and in marine straits*]{}, Stud. Appl. Math. [**80**]{} (1989), 1-23.

J. Ginibre and G. Velo, [*Proprietes de lissage et existence de solutions pour l’equation de Benjamin-Ono generalisee*]{}, C.R. Acad. Sci. Paris [**308**]{} (1989), 309-314.

N. Hayashi, K. Nakamitsu, and M. Tsutsumi, [*On solutions of the initial value problem for the nonlinear Schrodinger equation in one space dimension*]{}, Math. Z. [**192**]{} (1986), 637-650.

N. Hayashi, K. Nakamitsu, and M. Tsutsumi, [*On solutions of the initial value problem for the nonlinear Schrodinger equation*]{}, J. Funct. Anal. [**71**]{} (1987), 218-245.

N. Hayashi and T. Ozawa, [*Smoothing effect for some Schrodinger equations*]{}, J. Funct. Anal. [**85**]{} (1989), 307-348.

T. Kato, [*On the Cauchy problem for the (generalized) Korteweg-de Vries equation*]{}, Adv. in Math. Suppl. Studies: Studies in Appl. Math. [**8**]{} (1983), 93-128.

B.B. Kadomtsev & V.I. Petviashvili, [*On the stability of solitary waves in weakly dispersive media*]{}, Sov. Phys. Dokl. [**15**]{} (1970), 539-541.

C.E. Kenig, [*On the local and global well-posedness theory for the KP-I equation*]{}, Annales de l’Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis [**21**]{}, 6 (2004) 827-838.

C. Kenig, G. Ponce, and L. Vega, [*Oscillatory integrals and regularity of dispersive equations*]{}, Indiana Univ. Math. J. [**40**]{}, No. 1 (1991), 33-69.

S. Kruzhkov and A.V. Faminskii, [*Generalized solutions of the Cauchy problem for the Korteweg-deVries equation*]{}, Math. U.S.S.R. Sb. [**48**]{} (1984), 93-138.

J. Levandosky, [*Gain of regularity for the KP-II equation*]{}, Indiana Univ. Math. J. [**49**]{} (2000), 353-403.

J. Levandosky, [*Smoothing properties of nonlinear dispersive equations in two spatial dimensions*]{}, J. Differential Equations [**175**]{} (2001), 275-352.

L. Molinet, J.C. Saut, N. Tzvetkov, [*Global well-posedness for the KP-I equation on the background of a non-localized solution*]{}, Communications in Mathematical Physics [**272**]{}, 3 (2007) 775-810.

V.I. Petviashvili & V.V. Yan’kov, [*Solitons and turbulence*]{}, Rev. Plasma Phys., XIV (1989), 1-62.

G. Ponce, [*Regularity of solutions to nonlinear dispersive equations*]{}, J. Differential Equations [**78**]{} (1989), 122-135.

J.C. Saut, [*Remarks on the generalized Kadomtsev-Petviashvili equations*]{}, Ind. Univ. Math. J. [**42**]{} (1993), 1011-1026.

P.M. Santini, [*On the evolution of two-dimensional packets of water waves over an uneven bottom*]{}, Lett. Nuovo Cim. [**30**]{} (1981), 236-240.

S. Ukai, [*Local solutions of the Kadomtsev-Petviashvili equation*]{}, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. [**36**]{} (1989), 193-209.

O. Vera, [*Gain of regularity for a generalized coupled system of nonlinear evolution dispersive equations type,*]{} Ph. D. Thesis, UFRJ, Rio de Janeiro, Brazil(2001).

O. Vera, [*Gain of regularity for a Korteweg-de Vries-Kawahara type equation*]{}, Electron. J. Differential Equations [**71**]{} (2004), 1-24.

[^1]: Mathematics Department, Framingham State College, Framingham, MA 01701, jlevandosky@frc.mass.edu

[^2]: Departamento de Ingenier’[i]{}a Matem’[a]{}tica, Universidad de Concepci’[o]{}n, Casilla 160-C, Concepci’[o]{}n, Chile. mauricio@ing-mat.udec.cl

[^3]: Departamento de Matem’[a]{}tica, Universidad del B’[i]{}o-B’[i]{}o Collao 1202, Casilla 5-C, Concepci’[o]{}n, Chile, overa@ubiobio.cl
