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Global solutions of the Cauchy problem for a viscous polytropic ideal gas. (English) Zbl 0928.35134
The author considers the Cauchy problem for a viscous polytropic ideal gas in \(\mathbb{R}^n\) \((n=2\) or 3). Assuming that the initial density, velocity and temperature \((\rho_0,v_0, \theta_0)\) are close to \((\overline \rho,0, \overline \theta)\) in \(H^1\cap W^{1,\alpha} \times H^1\times H^1\) \((\overline \rho,\overline \theta\) constant, \(n<\alpha<{2n\over n-2})\), he obtains the existence and uniqueness of a global solution. The main ingredients of the proof are the usual energy arguments, the second law of thermodynamics and exponential decay properties of \(\rho-\overline\rho\). An important role in the analysis is played by the effective viscous flux which is a suitable combination of the divergence of the velocity and a pressure term. The result improves earlier work by Matsumura and Nishida and should also be compared to a recent result by D. Hoff [Arch. Ration. Mech. Anal. 139, No. 4, 303-354 (1997; Zbl 0904.76074)].

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