On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization

Abstract

We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta ,u,\theta )$, in a norm slightly stronger than $L^{2,\infty }(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $({\eta }^0,u^0,e^0)$ in a norm of $L^2(\Omega )\times H^{-1}(\Omega )\times H^{-1}(\Omega )$-type and also on the free terms in all the equations in some dual norms. Here $\eta$, $u$ and $\theta$ are the specific volume, velocity and absolute temperature as well as ${\eta }^0$, $u^0$ and $e^0$ are the initial specific volume, velocity and specific total energy, and $Q=\Omega \times (0,T)$. We also apply this result to the case of discontinuous rapidly oscillating, with the period $\epsilon$, initial data and free terms and derive an estimate $O(\epsilon )$ for the difference between the solutions to the Navier-Stokes equations and their Bakhvalov-Eglit two-scale homogenized version with averaged data.

30 pages