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Global solution and large-time behavior of the \(3D\) compressible Euler equations with damping. (English) Zbl 1259.35162

Summary: We construct the global unique solution to the compressible Euler equations with damping in \(R^{3}\). We assume the \(H^{3}\) norm of the initial data is small, but the higher order derivatives can be arbitrarily large. When the \(\Dot{H}^{-s}\) norm \((0\leqslant s< 3/2)\) or \(\dot{B}^{-s}_{2, \infty}\) norm \((0<s\leqslant 3/2)\) of the initial data is finite, by a regularity interpolation trick, we prove the optimal decay rates of the solution. As an immediate byproduct, the \(L^{p}\)-\(L^{2}(1\leqslant p\leqslant 2)\) type of the decay rates follow without requiring that the \(L^{p}\) norm of initial data is small.

MSC:

35Q31 Euler equations
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
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