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<p>This paper presents a comprehensive investigation of spherically symmetric solutions for the compressible Euler equations with damping in <i>N</i>-dimensional space. Through rigorous mathematical analysis, we establish that the spherically symmetric solutions exhibiting linear velocity dependence along the radial direction in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> can be characterized by a second-order nonlinear ordinary differential equation. By employing qualitative analysis techniques from planar dynamical systems theory, we systematically examine the phase-space trajectories of the corresponding dynamical system. This analytical approach enables us to identify and classify both blowup phenomena and global existence of spherically symmetric solutions for the considered compressible Euler equations.</p>

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Blowup and Global Spherically Symmetric Solutions to the Compressible Euler Equations with Damping: Dynamical Systems Approach

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月14日

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  • Guangxuan He,
  • Manwai Yuen,
  • Lijun Zhang

摘要

This paper presents a comprehensive investigation of spherically symmetric solutions for the compressible Euler equations with damping in N-dimensional space. Through rigorous mathematical analysis, we establish that the spherically symmetric solutions exhibiting linear velocity dependence along the radial direction in \(\mathbb {R}^N\) R N can be characterized by a second-order nonlinear ordinary differential equation. By employing qualitative analysis techniques from planar dynamical systems theory, we systematically examine the phase-space trajectories of the corresponding dynamical system. This analytical approach enables us to identify and classify both blowup phenomena and global existence of spherically symmetric solutions for the considered compressible Euler equations.