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Compressible Navier-Stokes equations with degenerate viscosities

We will discuss recent results on the construction of weak solutions for3D compressible Navier-Stokes equations with degenerate viscosities,including the construction of weak solutions for the viscous shallow waterequation. This was proposed as an open problem by P.-L. Lions in his bookon fluid mechanics in the late 90'. It has been solved, in collaborationwith Alexis Vasseur,in our recent joint work. We will also mention someresults on the energy conservation for the Navier-Stokes equations.