Solutions classiques globales des équations d'Euler pour un fluide parfait compressible

Serre, Denis

doi:10.5802/aif.1563

Serre, Denis

Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 139-153.

Résumé
Abstract

Soit $ρ$ , $u$ , $e$ , $S$ et $p$ les variables usuelles qui décrivent l’état d’un fluide en coordonnées eulériennes. Le domaine physique occupé par le fluide est a priori $ℝ^{d}$ tout entier, mais $ρ$ peut être nul en dehors d’un compact $K (t)$ . On choisit l’équation d’état d’un gaz parfait, $p = (γ - 1) ρ e$ , où $γ \in [1, 1 + 2 / d]$ est une constante. Le cas $γ = 1 + 2 / d$ est celui du gaz mono-atomique.

Dans la limite $ρ \to 0$ , les collisions sont rares et on est tenté d’approcher le mouvement des particules par un mouvement rectiligne uniforme : le champ de vitesse obéit alors à $v_{t} + (v \cdot \nabla) v = 0$ . Si de plus $v (0, x) = A_{0} x$ , où $A_{0} \in M_{d} (ℝ)$ n’a pas de valeur propre réelle négative, $v$ est défini pour tout $t \geq 0 : v (t, x) = A (t) x$ , ce qu’on note $u_{A (t)}$ .

On montre ici que, pour une condition initiale $ρ_{0}$ , $u_{0}$ , $S_{0}$ telle que $ρ_{0}^{(γ - 1) / 2}$ , $u_{0} - u_{A_{0}}$ , $S_{0} - \bar{S}$ ( $\bar{S}$ étant une constante) soient petits dans $H^{m} (ℝ^{d})$ ( $m > 1 + d / 2$ ), le problème de Cauchy admet une solution classique pour tout $t \geq 0$ . Si de plus $A_{0}$ n’a aucune valeur propre réelle (dans ce cas, $d$ est pair), l’existence a lieu pour tout $t \in ℝ$ . Enfin, pour un gaz mono-atomique, on donne une description précise du comportement asymptotique en temps.

Let $ρ$ , $u$ , $e$ , $S$ , $p$ the usual variables describing the state of a fluid in an eulerian frame. The underlying physical space is $ℝ^{d}$ , $d \geq 1$ . We restrict to the perfect gas law: $p = (γ - 1) ρ e$ , where $γ \in [1, 1 + 2 / d]$ is a constant. In the formal limit $ρ \to 0$ (rarefied gases), the particles evolve freely with a uniform motion; if the initial velocity field is linear (say $u_{0} (x) = A_{0} x$ ), then it remains so, with $A^{'} (t) = - A (t)^{2}$ , and it is defined for every positive time, provided $A_{0}$ does not have a non-positive real eigenvalue. Let $u_{A}$ be this field. The purpose of this paper is to prove that, if the initial data $(ρ_{0}, u_{0}, S_{0})$ is close to $(0, u_{A}, \bar{S})$ , with $\bar{S}$ a constant, then the Cauchy problem admits a (unique) smooth solution defined for all $t \geq 0$ . In the mono-atomic case ( $γ = 1 + 2 / d$ ), we give an accurate description of the asymptotic behaviour. All these results are especially designed for finite mass flows. The above-mentioned closeness relies as usual to the space $H^{m} (ℝ^{d})$ with $m > 1 + d / 2$ .

In even space dimension (say $d = 2$ ), our result shows the existence of non-trivial smooth flows defined for all times $t \in ℝ$ .

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.1563

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     title = {Solutions classiques globales des \'equations {d'Euler} pour un fluide parfait compressible},
     journal = {Annales de l'Institut Fourier},
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     number = {1},
     year = {1997},
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Serre, Denis. Solutions classiques globales des équations d'Euler pour un fluide parfait compressible. Annales de l'Institut Fourier, Tome 47 (1997) no. 1, pp. 139-153. doi : 10.5802/aif.1563. https://www.numdam.org/articles/10.5802/aif.1563/

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