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Solutions classiques globales des équations d’Euler pour un fluide parfait compressible. (Global smooth solutions for the Euler equations of a perfect compressible fluid.). (French) Zbl 0864.35069
Summary: Let \(\rho\), \(u\), \(e\), \(S\), \(p\) be the usual variables describing the state of a fluid in an Eulerian frame. The underlying physical space is \({\mathbb{R}}^d\), \(d\geq 1\). We restrict ourselves to the perfect gas law: \(p=(\gamma-1)\rho e\), where \(\gamma\in[1,1+2/ d]\) is a constant. In the formal limit \(\rho\to 0\) (rarefied gases), the particles evolve freely with a uniform motion; if the initial velocity field is linear (say \(u_0(x)=A_0x\)), 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 \((\rho_0,u_0,S_0)\) are 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 (\(\gamma=1+2/ d\)), 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 \(H^m({\mathbb{R}}^d)\) with \(m>1+d/ 2\). In an even space dimension (say \(d=2\)), our result shows the existence of non-trivial smooth flows defined for all times \(t\in{\mathbb{R}}\).

35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35Q35 PDEs in connection with fluid mechanics
76N15 Gas dynamics (general theory)
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[1] S. ALINHAC, Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux, Inventiones Mathematicae, vol 111 (1993), 627-678. · Zbl 0798.35129
[2] J.-Y. CHEMIN, Dynamique des gaz à masse totale finie, Asymptotic Analysis, vol 3 (1990), 215-220. · Zbl 0708.76110
[3] L. GARDING, Problèmes de Cauchy pour LES systèmes quasi-linéaires d’ordre un strictement hyperboliques, in Les équations aux dérivées partielles, Colloques internationaux du CNRS, vol 117, 33-40, Paris 1963. · Zbl 0239.35013
[4] P.D. LAX, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., vol 5 (1964), 611-613. · Zbl 0135.15101
[5] P.-L. LIONS, Existence globale de solutions pour LES équations de Navier-Stokes compressibles isentropiques, Comptes Rendus Acad. Sci. Paris, vol 316 (1993), 1335-1340 et : Compacité des solutions des équations de Navier-Stokes compressibles isentropiques, Comptes Rendus Acad. Sci. Paris, vol 317 (1993), 115-120. · Zbl 0778.76086
[6] T. MAKINO, S. UKAI et S. KAWASHIMA, Sur la solution à support compact de l’équation d’Euler compressible, Japan J. Appl. Math., vol 3 (1986), 249-257. · Zbl 0637.76065
[7] A. MAJDA, Compressible fluid flows and systems of conservation laws in several space variables, Appl. Math. Sci. Ser., vol 53. Springer-Verlag, Berlin, 1983. · Zbl 0537.76001
[8] D. SERRE, Systèmes de lois de conservation, Diderot, Paris, 1996.
[9] T. SIDERIS, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., vol 101 (1985), 475-485. · Zbl 0606.76088
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