Convergence to diffusion waves for solutions of Euler equations with time-depending damping on quadrant

TL;DR

This paper investigates the long-term behavior of solutions to Euler equations with time-dependent damping on a quadrant, proving convergence to nonlinear diffusion waves and extending previous results to more general initial conditions.

Contribution

It establishes the global existence and asymptotic convergence of solutions with time-dependent damping, generalizing prior work with constant damping coefficients.

Findings

Solutions tend to nonlinear diffusion waves over time.

Global smooth solutions exist under specified boundary conditions.

Convergence rates differ from previous Cauchy problem results.

Abstract

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\frac{\alpha}{(1+t)^\lambda} u, \end{equation} with null-Dirichlet boundary condition or 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 Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts 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.