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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)
##### Keywords:
gas dynamics; nonlinear hyperbolic equations
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##### References:
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