Section: New Results

Asymptotic analysis for fluid mechanics

In [28], Ingrid Lacroix-Violet and Alexis Vasseur present the construction of global weak solutions to the quantum Navier-Stokes equation, for any initial value with bounded energy and entropy. The construction is uniform with respect to the Planck constant. This allows to perform the semi-classical limit to the associated compressible Navier-Stokes equation. One of the difficulty of the problem is to deal with the degenerate viscosity, together with the lack of integrability on the velocity. The method is based on the construction of weak solutions that are renormalized in the velocity variable. The existence and stability of these solutions do not need the Mellet-Vasseur inequality.

In [44], the main objective is to generalize to the Navier-Stokes-Korteweg (with density dependent viscosities satisfying the BD relation) and Euler-Korteweg systems a recent relative entropy proposed in [65]. As a concrete application, this helps to justify mathematically the convergence between global weak solutions of the quantum Navier-Stokes system and dissipative solutions of the quantum Euler system when the viscosity coefficient tends to zero. Our results are based on the fact that Euler-Korteweg systems and corresponding Navier–Stokes-Korteweg systems can be reformulated through an augmented system. As a by-product of our analysis, we show that this augmented formulation helps to define relative entropy estimates for the Euler-Korteweg systems in a simplest way and with less hypothesis compared to recent works [74], [80].

In [27], Pierre-Emmanuel Jabin and Thomas Rey investigate the behavior of granular gases in the limit of small Knudsen number, that is, very frequent collisions. They deal with the strongly inelastic case in one dimension of space and velocity. They are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit. A more general result is also presented, which can apply to a large class of energy-dissipative kinetic equations.