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
Alexander Zlotnik
TL;DR
The paper advances the theory of the 1D compressible Navier–Stokes system by proving Lipschitz-type dependence of global weak solutions $(\eta,u,\theta)$ on initial data and forcing in dual-type norms for large discontinuous data and nonhomogeneous boundaries. It generalizes previous data-dependence results to include perturbations in mass, momentum, and energy balances with Eulerian-coordinate dependent forcing, applicable to three boundary types, and derives Hölder-type corollaries in stronger norms. The authors then apply this continuity framework to a homogenization setting with discontinuous rapidly oscillating data, establishing an $O(\varepsilon)$ convergence rate to the Bakhvalov–Eglit two-scale homogenized model with averaged data, along with finer $O(\varepsilon^{1/2})$ and $O(\varepsilon^{1/4})$ bounds in stronger norms. Overall, the work provides both rigorous data-sensitivity results for weak solutions and practical homogenization error estimates for compressible flow models.
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 $(η,u,θ)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(η^0,u^0,e^0)$ in a norm of $L^2(Ω)\times H^{-1}(Ω)\times H^{-1}(Ω)$-type and also on the free terms in all the equations in some dual norms. Here $η$, $u$ and $θ$ are the specific volume, velocity and absolute temperature as well as $η^0$, $u^0$ and $e^0$ are the initial specific volume, velocity and specific total energy, and $Q=Ω\times (0,T)$. 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.