Sharp critical thresholds in a hyperbolic system with relaxation

TL;DR

This paper investigates a hyperbolic Eulerian system with relaxation, identifying critical initial data thresholds that determine whether solutions remain regular or blow up, highlighting the system's dynamic hyperbolicity transition.

Contribution

It introduces a novel analysis of critical thresholds in a hyperbolic system with relaxation, including bounds and conditions for hyperbolicity and solution behavior.

Findings

Critical thresholds separate global regularity and blowup.

Bounds on density depend on velocity for certain relaxation types.

System transitions between strictly and weakly hyperbolic regimes.

Abstract

We propose and study a one-dimensional $2\times 2$ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.