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 U0, U0′∈Hk and U, U′ ∈ C (0, T; Hk) denote initial data and corresponding solutions, respectively. From the point of view of dynamical systems, a very basic problem is to prove that U′ converges to U in С (0, Τ; Hk) if U0′ converges to U0 in Hk; this is proved in theorem 1.2 below. It must be pointed out that convergence in C(0, T; Hk − ε) and in L∞(0, T; Hk) weak-* (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 ρ′ (p, S) approaches ρ (p, S) in Ck then U′ approaches U in the norm C (0, T; Hk). In particular, small perturbations in the law of state generate small perturbations in the trajectory of the solution, with respect to the right metric.