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<p>This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for 1D compressible Euler equations with damping of time and space dependent coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha (x,t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which models the compressible flow through porous media. We prove that the solutions to this system globally exist and converge to the diffusion waves, which are the self-similar solutions to the corresponding nonlinear parabolic equation given by Darcy’s law. The optimal convergence rates are also obtained. The proof is accomplished by virtue of energy estimates.</p>

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Asymptotic Behavior of Solutions to Compressible Euler Equations with Time and Space Dependent Damping

  • 期刊论文

  • 发表日期：

    2025年12月5日

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  • Nangao Zhang

摘要

This paper is concerned with the asymptotic behavior of solutions to the Cauchy problem for 1D compressible Euler equations with damping of time and space dependent coefficient \(\alpha (x,t)\) α ( x , t ) , which models the compressible flow through porous media. We prove that the solutions to this system globally exist and converge to the diffusion waves, which are the self-similar solutions to the corresponding nonlinear parabolic equation given by Darcy’s law. The optimal convergence rates are also obtained. The proof is accomplished by virtue of energy estimates.