Linear transport equations with initial values in Sobolev spaces and application to the Navier-Stokes equations

The purpose of this note is to present new results on linear transport equations with initial values in W1,r(Ω) spaces, and coefficients in Ws+1,p(Ω)N, where Ω is a bounded open subset of RN (N ≥ 2) and $sp=N$. As an application, we also give refined regularity and uniqueness results for weak solutions of the two-dimensional density-dependent incompressible Navier-Stokes equations.