Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow

TL;DR

This paper investigates the long-term stability of multi-dimensional plane shocks in isothermal flow, revealing their global instability and providing lifespan estimates similar to nonlinear wave equations.

Contribution

It demonstrates the non-existence of stable fan-shaped wave solutions in multi-dimensional isothermal Euler flows and establishes sharp lifespan estimates.

Findings

Multi-dimensional plane shocks are globally unstable.

Fan-shaped wave solutions do not exist for these flows.

Lifespan estimates match those of nonlinear wave equations.

Abstract

In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. Non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity and which is a perturbation of plane waves. Therefore, unlike the one-dimensional case, the multi-dimensional plane shocks are not stable globally. What is more, the sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.