An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation, is given by, ( w , ∂ ∂ u ∂ ∂ t ) = − − ( w , ( u ⋅ ⋅ ∇ ∇ ) u ) − − ν ν ( ∇ ∇ w : ∇ ∇ u ) + ( w , f S )