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

TL;DR

This paper establishes Lipschitz continuous dependence of 1D compressible Navier-Stokes solutions on initial data and free terms in weaker norms, and provides an $O( ext{epsilon})$ error estimate for homogenization with oscillating initial data.

Contribution

It proves continuous dependence of weak solutions in weaker norms and derives an $O( ext{epsilon})$ homogenization error estimate for oscillatory initial data.

Findings

Lipschitz-type dependence in weaker norms

Error estimate of $O( ext{epsilon})$ for homogenization

Applicability to discontinuous oscillating data

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 $\varepsilon$, initial data and free terms and derive an estimate $O(\varepsilon)$ for the difference between the solutions to the Navier-Stokes equations and their Bakhvalov-Eglit two-scale homogenized version with averaged data.