On the isentropic compressible Navier-Stokes equation

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We consider the compressible Navier-Stokes equation with density dependent viscosity coefficients, focusing on the case where those coefficients vanish on vacuum. We prove the stability of weak solutions both in the torus and in the whole space in dimension 2 and 3. The pressure is given by p=rhoᵍamma, and our result holds for any gamma>1. In particular, we obtain the stability of weak solutions of the Saint-Venant model for shallow water.