Global weak solutions to the compressible quantum Navier–Stokes equation and its semi-classical limit

Abstract

This paper is dedicated to 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. Our 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.

Global Existence of Finite Energy Weak Solutions of Quantum Navier–Stokes Equations

“…The following is the analogue result on the whole space to [7,25] for κ > 0 and [29,35] for κ = 0 respectively. Theorem 2.4.…”

“…In this section, we discuss the existence of a sequence of weak solutions to the system (1.1) on T d n with initial data ( ρ n 0 , ρ n 0 u n 0 ) provided by Lemma 3.2. In [25], the authors show global existence of weak solutions to (1.1) posed on [0, T ) × T d n for γ > 1, ν > 0 and κ ≥ 0 complemented with initial data of finite energy. The construction of weak solutions proceeds in several steps:…”

“…The solutions obtained in [7] can be shown to be finite energy weak solutions to (1.1) on T d n in the sense of Definition 2.1, namely the solutions provided in [7] satisfy (2.1) and (2.2) with C = 1, see Appendix A of [1]. Following the arguments in [25] and in the Appendix A of [1], it can be checked that the solutions of [25] also enter the class of finite energy weak solution in the sense of Definition 2.1. We define…”

“…It remains to show that the truncated finite energy weak solution (ρ, u) provided by Theorem 4.1 is a finite energy weak solution to (1.1) with (1.2). We proceed as in [25]. The proof uses a local argument for which the far-field plays a minor role, we omit the details.…”

Remarks on the derivation of finite energy weak solutions to the QHD system

“…For this reason in order to deal with finite energy weak solutions to (1.1), it is more convenient to consider the unknowns ( √ ρ, Λ), which define the hydrodynamic quantities by ρ = ( √ ρ) 2 , J = √ ρΛ, see Definition 15 below for more details. The lack of suitable a priori bounds prevents the study of solutions to (1.1) by using compactness arguments like it is done for viscous systems, see for example [8,7,51,58] where a viscous counterpart of system (1.1) is considered, see also [9,10] where a similar system is studied by using a suitable truncation argument. On the contrary, for the QHD system most of the existing results in the literature are perturbative [47,53,41,42].…”