Global existence of weak solutions to the compressible quantum Navier-Stokes equations with degenerate viscosity

Abstract

We study the compressible quantum Navier-Stokes (QNS) equations with degenerate viscosity in the three dimensional periodic domains. On the one hand, we consider QNS with additional damping terms. Motivated by the recent works of Li and Xin [e-print arXiv:1504.06826] and P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)], we construct a suitable approximate system which has smooth solutions satisfying the energy inequality and the BD entropy estimate. Using this system, we obtain the global existence of weak solutions to the compressible QNS equations with damping terms for large initial data. Moreover, we obtain some new a priori estimates, which can avoid using the assumption that the gradient of the velocity is a well-defined function, which was indeed used directly in the work of Vasseur and Yu [SIAM J. Math. Anal. 48, 1489–1511 (2016); Invent. Math. 206, 935–974 (2016)]. On the other hand, in the absence of damping terms, we also prove the global existence of weak solutions to the compressible QNS equations without the lower bound assumption on the dispersive coefficient, which improves the previous result of P. Antonelli and S. Spirito [Arch. Ration. Mech. Anal. 225, 1161–1199 (2017)].

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

“…(1) The weak solutions to an auxiliary system including drag forces provided in [35] satisfy a truncated formulation of the equations. By a different approach, namely constructing smooth approximate weak solutions to (1.1), the existence of weak solutions to (1.1) posed on T d n satisfying properties (i)-(iii) has also been obtained in [7,33], following the strategy of [29]. These results require additional restrictions on γ, κ.…”

“…The global existence of weak solutions in two and three dimensions has been obtained in [35] and [29] with vanishing boundary conditions at infinity. Concerning the quantum Navier-Stokes system, the existence of finite energy weak solutions on the torus T d , for d = 2, 3 has been proved in [7] and [25], see also [6,33]. Global existence of weak solutions to the isothermal quantum Navier-Stokes equations, i.e.…”

Global existence of weak solutions to the Navier–Stokes–Korteweg equations

“…This was noticed in [34] for the one-dimensional problem and in [25] for spherically symmetric solutions; see also [35,Rem. 1.5] and the discussion in [40]. Regarding the existence of weak solutions, important progress was made at around the same time in [35] and [43].…”

“…This, together with the bounds provided by the BD entropy, yields sufficient compactness for the unknowns. In [4,5] this strategy was adopted in order to prove global existence of finite energy weak solutions to the quantum Navier-Stokes system with the standard notion of weak solutions;, see also [14,33,40]. Let us stress that for system (1.1)-(1.2) the relation (1.9) does not hold true and consequently it is not possible to derive a Mellet-Vasseur-type estimate.…”

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].…”