Seminar Analysis and Mathematical Physics: Lars Eric Hientzsch (Institut Fourier, University of Grenoble Alpes)

Existence of weak solutions and incompressible limit for quantum Navier-Stokes equations

The quantum Navier-Stokes (QNS) equations describe a compressible fluid including a degenerate density dependent viscosity and a dispersive tensor accounting for capillarity effects. The system can be seen as viscous correction of the Quantum Hydrodynamics (QHD) arising e.g. as prototype model in the description of superfluidity. We consider the (QNS) system on the whole space with non-trivial farfield behaviour providing the suitable framework to study coherent structures and the incompressible limit.

First, we prove global existence of finite energy weak solutions (FEWS) in dimension two and three. To compensate for the lack of control of the velocity field around vacuum regions, we construct approximate solutions to a truncated formulation of (QNS) on a sequence of invading domains. Suitable compactness properties are inferred from the Bresch-Desjadins entropy estimates. This is joint work with P. Antonelli and S. Spirito.

Second, we address the low Mach number limit for FEWS to the (QNS) system (in collaboration with P. Antonelli and P. Marcati). The main novelty is a precise analysis of the acoustic dispersion altered by the presence of the dispersive capillarity tensor. The linearised system is governed by the Bogoliubov dispersion relation. The desired decay of the acoustic part follows from refined Strichartz estimates. 

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