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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 \(t\in{\mathbb{R}}\).

MSC:
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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