# A variant to the Stokes system and Navier-Stokes equation

The linearization of the Navier-Stokes equation (in a smooth bounded domain in dimensions 2 or 3) is the following non-stationary Stokes system $$v_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$ whose $$W_p^{2,1}$$-estimates are well known due to the works of V.A. Solonnikov.

A variant of the Stokes system might be $$v_t+c(x)\nabla p=\Delta v+f,~\nabla\cdot v=g,$$ where the coefficient in front of the pressure $$p$$ depends also on space variable. I wonder if there is a name for the above variant. Does there exist the similar $$W_p^{2,1}$$-estimates as the standard Stokes system if we assume the coefficient $$c(x)$$ is sufficiently smooth?