The initial-boundary value problem for the non-barotropic compressible Euler equations: structural-stability and data dependence

Abstract

We study the initial-boundary value problem for general compressible inviscid fluids. Let ${\mathrm{U}}_0,{\mathrm{U}}_0^{\prime }\in {\mathrm{H}}^k$ and $U,U^{\prime }\in C(0,T;{\mathrm{H}}^k)$ denote initial data and corresponding solutions, respectively. From the point of view of dynamical systems, a very basic problem is to prove that ${\mathrm{U}}^{\prime }$ converges to $\mathrm{U}$ in $С(0,T;{\mathrm{H}}^k)$ if ${\mathrm{U}}_0^{\prime }$ converges to ${\mathrm{U}}_0$ in ${\mathrm{H}}^k$; this is proved in Theorem 1.2 below. It must be pointed out that convergence in $C(0,T;{\mathrm{H}}^{k-\epsilon })$ and in $L^{\infty }(0,T;{\mathrm{H}}^k)$ weak-$\ast$ (easy consequences of the a priori estimates used to prove the existence theorem) have minor significance as part of the mathematical theory. We also show (Theorem 1.3) that if ${\rho }^{\prime }(p,S)$ approaches $\rho (p,S)$ in $C^k$ then ${\mathrm{U}}^{\prime }$ approaches $\mathrm{U}$ in the norm $C(0,T;{\mathrm{H}}^k)$. In particular, small perturbations in the law of state generate small perturbations in the trajectory of the solution, with respect to the right metric.

Résumé

Nous étudions le problème mixte pour des fluides non visqueux, dans le cas général. Soient ${\mathrm{U}}_0,{\mathrm{U}}_0^{\prime }\in {\mathrm{H}}^k$ et $U,U^{\prime }\in C(0,T;{\mathrm{H}}^k)$ respectivement les données initiales et les solutions correspondantes. Du point de vue des systèmes dynamiques, un problème essentiel consiste à démontrer que ${\mathrm{U}}^{\prime }$ converge vers $\mathrm{U}$ dans $С(0,T;{\mathrm{H}}^k)$ lorsque ${\mathrm{U}}_0^{\prime }$ converge vers ${\mathrm{U}}_0$ dans ${\mathrm{H}}^k$; on démontre ce résultat dans le Théorème 1.2. En plus, on démontre (Théorème 1.3) que si $\rho \prime (p,S)$ converge vers $\rho (p,S)$ dans $C^k$ alors $\mathrm{U}\prime$ converge vers $\mathrm{U}$ dans $С(0,T;{\mathrm{H}}^k)$.