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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?

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