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This paper is dedicated to the construction of global weak solutions to thequantum Navier-Stokes equation, for any initial value with bounded energy andentropy. The construction is uniform with respect to the Planck constant. Thisallows to perform the semi-classical limit to the associated compressibleNavier-Stokes equation. One of the difficulty of the problem is to deal withthe degenerate viscosity, together with the lack of integrability on thevelocity. Our method is based on the construction of weak solutions that arerenormalized in the velocity variable. The existence, and stability of thesesolutions do not need the Mellet-Vasseur inequality.

Global weak solutions to the compressible quantum navier-stokes equation and its semi-classical limit