Paradoxes and challenges in the modeling of the viscosity for compressible viscous fluids
Abstract:
The classical derivation of the viscosity coefficients, for a viscous fluids, usually predicts a dependence on the temperature of the fluids only. However, when considering compressible fluids, this derivation leads to paradoxes associated to the possible presence of vacuum regions. When considering two disjoint bubbles of fluid, for instance, the associated compressible Navier-Stokes equation will predict unphysical long range interactions between those two bubbles. At the mathematical level, it implies the non uniqueness of the solutions, and unphysical scenarios for blow-ups in finite time. One way to avoid these problems consists in adding a dependence of the viscosities with respect to the density of the fluid. This family of models can be rigorously justified in certain situations, as for the viscous shallow water equation for instance. These models display unexpected beautiful structures, but carry their own mathematical challenges, due to the degeneracy of the viscosities close to the vacuum. We will discuss recent breakthroughs which have been accomplished recently to develop the mathematical grounding of these models (as, for instance, Vasseur-Yu [Inventiones mathematicae (2016) and arXiv:1501.06803 (2015)] and Li-Xin [arXiv:1504.06826 (2015)]).