JournalsaihpcVol. 11, No. 3pp. 297–311

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

  • H. Beirão da veiga

    Accademia Nazionale dei Lincei, Centro Linceo Interdisciplinare “B. Segre” via della Lungara, 10 Rome, Italy
The initial-boundary value problem for the non-barotropic compressible Euler equations: structural-stability and data dependence cover
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Abstract

We study the initial-boundary value problem for general compressible inviscid fluids. Let U0, U0Hk\mathrm{U}_{0}^{′} \in \mathrm{H}^{k} 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\mathrm{U}_{0}^{′} 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.

Résumé

Nous étudions le problème mixte pour des fluides non visqueux, dans le cas général. Soient U0, U0Hk\mathrm{U}_{0}^{′} \in \mathrm{H}^{k} et U, U′ ∈ C (0, T; Hk) respectivement les données initiales et les solutions correspondantes. Du point de vue des systèmes dynamiques, un problème essentiel consiste à démontrer que U′ converge vers U dans С (0, T; Hk) lorsque U0\mathrm{U}_{0}^{′} converge vers U0 dans Hk; on démontre ce résultat dans le théorème 1.2. En plus, on démontre (théorème 1.3) que si ρ′ (p, S) converge vers ρ (p, S) dans Ck alors U′ converge vers U dans С (0, T; Hk).

Cite this article

H. Beirão da veiga, The initial-boundary value problem for the non-barotropic compressible Euler equations: structural-stability and data dependence. Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 3, pp. 297–311

DOI 10.1016/S0294-1449(16)30186-X