Global existence versus blow-up for multi-d hyperbolized compressible Navier-Stokes equations

TL;DR

This paper investigates the global existence and blow-up phenomena for a modified compressible Navier-Stokes system incorporating hyperbolic heat conduction and Maxwellian flow, revealing conditions for stability and finite-time singularity.

Contribution

It introduces a new model replacing Fourier's law with Cattaneo's law and classical flow with Maxwell flow, analyzing its well-posedness and blow-up behavior.

Findings

Existence of a physical entropy for the new model.

Global solutions for small initial data in certain cases.

Finite-time blow-up for large initial data.

Abstract

We consider the non-isentropic compressible Navier-Stokes equations in two or three space dimensions for which the heat conduction of Fourier's law is replaced by Cattaneo's law and the classical Newtonian flow is replaced by a revised Maxwell flow. We show that a physical entropy exists for this new model. For two special cases, we show the global well-posedness of solutions with small initial data and the blow-up of solutions in finite time for a class of large initial data. Moreover, for vanishing relaxation parameters, the solutions (if it exists) are shown to converge to solutions of the classical system.