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Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant. (English) Zbl 1412.35238

Summary: This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant \((x,t) \in\mathbb{R}^+\times\mathbb{R}^+\), \[ \partial_tv - \partial_xu = 0,\quad\partial_tu + \partial_xp\left( v \right) = - \frac{\alpha}{(1+t)^\lambda}u, \] with the null-Dirichlet boundary condition or the null-Neumann boundary condition on \(u\). We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by K. Nishihara and T. Yang [J. Differ. Equations 156, No. 2, 439–458 (1999; Zbl 0933.35121)], and M. Jiang and C. Zhu [Discrete Contin. Dyn. Syst. 23, No. 3, 887–918 (2009; Zbl 1170.35058)], we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the fact that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.

MSC:

35Q31 Euler equations
35L65 Hyperbolic conservation laws
76N15 Gas dynamics (general theory)
35B45 A priori estimates in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

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