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Global smooth solutions to Euler equations for a perfect gas. (English) Zbl 0930.35134
For the Euler equations which describe the perfect gas with equation of state $$p= (\gamma-1)\rho^\gamma$$, the author studies the Cauchy problem with initial data $$u(x,0)= u_0(x)$$, $$\rho_0(x, 0)= \rho_0(x)$$ ($$p$$ stands for pressure, $$\rho$$ for density, $$x\in \mathbb{R}^d$$, $$d\geq 1$$). Under the assumptions
(I) $$\rho^{(\gamma- 1)/2}_0$$ is small enough in $$H^m(\mathbb{R}^d)$$;
(II) $$D^2u_0\in H^{m- 1}(\mathbb{R}^d)$$ and $$Du_0\in L^\infty(\mathbb{R}^d)$$;
(III) there exists $$\delta>0$$ such that for all $$x\in \mathbb{R}^d$$ the distance between $$\mathbb{R}_-$$ and spectrum of $$Du_0(x)$$ is greater than $$\delta$$;
(IV) $$\rho_0$$ has a compact support, the author proves the existence and uniqueness of global smooth solution $$(\rho,u)$$ such that $(\rho^{(\gamma- 1)/2}_0, u-\overline u)\in C^j([0, \infty[; H^{m- j}(\mathbb{R}^d))\quad\text{for }j\in \{0,1\}.$ The proof is based on energy estimates.
Reviewer: O.Titow (Berlin)

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
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