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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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