Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum
This paper is devoted to one-dimensional compressible Navier-Stokes equations for isentropic flow, where the pressure law is that of polytropic gases (\(P(\rho) = A(\rho^\gamma)\)), and where the viscosity coefficient in the equation for velocity field depends on density. Such a dependence is due to the derivation of Navier-Stokes equations from Boltzmann equation through Chapman-Enskog expansion to the second order (in which the visocity is not constant, but depends on the temperature). In the case of isentropic flow that dependence can be translated to a density dependence. The initial data is supposed to be compactly supported. That means that there is an interface between the gas and vacuum. Of course, since the viscosity depends on the density, that means that the viscosity coefficient degenerates in vacuum. A local existence result in that framework was obtained previously by \textit{T. Yang} and \textit{H. Zhao} [J. Differ. Equations 184, No. 1, 163--184 (2002; Zbl 1003.76073)]. The goal here is to derive new a priori estimates (in weighted spaces), enabling the authors to prove global existence of weak solutions, in the case of compactly supported density (in Eulerian coordinates) as well as in the case of an infinite support but with finite mass. A non-global existence theorem for smooth solutions is also proved for compactly supported data (an open problem for the full non-isentropic Navier-Stokes equation). To prove the results, the authors translate the problem into Lagrangian coordinates, and the theorems are stated in this formulation. The proof of the results is first sketched, omitting some technicalities which are proved at the end of the paper. The main lemma consists in a lower bound on the density function.
scientific article; zbMATH DE number 1942702