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)].