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Global existence and asymptotic behavior of solutions to the Euler equations with time-dependent damping. (English) Zbl 07441233

Summary: We study the isentropic Euler equations with time-dependent damping, given by \(\frac{\mu}{(1+t)^{\lambda}} \rho\, u\). Here, \(\lambda\) and \(\mu\) are two non-negative constants to describe the decay rate of damping with respect to time. We will investigate the global existence and asymptotic behavior of small data solutions to the Euler equations when \(0<\lambda<1,\, 0<\mu\) in multi-dimensions \(n\geq 1\). Our strategy of proving the global existence is to convert the Euler system to a time-dependent damped wave equation and use a kind of weighted energy estimate. Investigation to the asymptotic behavior of the solution is based on the detailed analysis to the fundamental solutions of the corresponding linear damped wave equation and it coincides with that of standard results if \(\lambda\) deduces to zero.

MSC:

35Q31 Euler equations
35L70 Second-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
76N15 Gas dynamics (general theory)
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