It is usually stated by equating to zero the total time derivative, plus the force dotted with the velocity gradient, minus the collision operator, all operating on the distribution function of a fluid.
Derived in full conservation form by Axford, this PDE can be obtained by ensemble-averaging moments of the Klimontovich equation, which represents as delta functions the objects that this equation represents as a continuum. Its homogeneous version is the Vlasov equation, which has no collision-operator term. The continuum equation is its zeroth velocity moment; higher moments yield momentum relations like the Navier-Stokes equation. It is usually stated by equating to zero the total time derivative, plus the force dotted with the velocity gradient, minus the collision operator, all operating on the distribution function of a fluid. FTP, identify this equation for the distribution function of a fluid, named for an Austrian physicist.
Derived in full conservation form by Axford, this PDE can be obtained by ensemble-averaging moments of the Klimontovich equation, which represents as delta functions the objects that this equation represents as a continuum.
Boltzmann equation (prompt on "Vlasov equation" before it is mentioned)