Singularity formation of hydromagnetic waves in cold plasma

TL;DR

This paper investigates the conditions under which smooth solutions to a cold plasma fluid model develop singularities in finite time, revealing that even initially smooth states can lead to blow-up due to specific dynamics.

Contribution

It introduces new sufficient conditions for $C^1$ blow-up in a magnetized cold plasma model, including cases with initially zero velocity gradient, and reduces the problem to analyzing a second-order ODE.

Findings

Smooth solutions can break down in finite time without initial velocity gradient.

Density and velocity gradient become unbounded near blow-up time.

Lagrangian formulation simplifies singularity analysis to a second-order ODE.

Abstract

We study $C^1$ blow-up of the compressible fluid model introduced by Gardner and Morikawa, which describes the dynamics of a magnetized cold plasma. We propose sufficient conditions that lead to $C^1$ blow-up. In particular, we find that smooth solutions can break down in finite time even if the gradient of initial velocity is identically zero. The density and the gradient of the velocity become unbounded as time approaches the lifespan of the smooth solution. The Lagrangian formulation reduces the singularity formation problem to finding a zero of the associated second-order ODE.