Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions

Abstract

Compressible Euler equations with space-dependent damping in high dimensions Rn(n=2,3){{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3) are considered in this article. Assuming that the small initial velocity and small perturbation of the initial density have compact support, we establish finite-time blow-up results for the Euler system, by combining energy estimate and new test functions constructed by the solutions of the following linear elliptic partial differential equations system: −G1(x)+∇⋅G2→(x)=0,−G2→(x)+∇G1(x)=μG2→(x)(1+∣x∣)λ.\left\{\begin{array}{l}-{G}_{1}\left(x)+\nabla \cdot \overrightarrow{{G}_{2}}\left(x)=0,\\ -\overrightarrow{{G}_{2}}\left(x)+\nabla {G}_{1}\left(x)=\frac{\mu \overrightarrow{{G}_{2}}\left(x)}{{(1+| x| )}^{\lambda }}.\end{array}\right. This result generalizes the one in the literature from 1−D1-D to high dimension Rn(n=2,3){{\bf{R}}}^{n}\hspace{0.33em}\hspace{0.33em}\left(n=2,3).

Keywords

• compressible euler equations
• damping
• blow-up
• lifespan
• test function method
• 35q31
• 35l65
• 35l67
• 76n15