Global Spherically Symmetric Solutions of the Multidimensional Full Compressible Navier-Stokes Equations with Large Data

TL;DR

This paper proves the existence of global solutions to the multidimensional compressible Navier-Stokes equations with large, discontinuous, spherically symmetric initial data, allowing for cavitation and vacuum regions.

Contribution

It introduces a novel approximation approach using finite annular regions to establish global solutions with large, discontinuous initial data in multidimensional settings.

Findings

Global solutions exist for large, discontinuous initial data

Density remains bounded away from vacuum in regions away from cavitation

Solutions exhibit H"older continuity in velocity and internal energy

Abstract

We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier-Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the possible vacuum, the velocity and specific internal energy are H\"older continuous, and the density has a uniform upper bound. To achieve these, our main strategy is to regard the Cauchy problem as the limit of a series of carefully designed initial-boundary value problems that are formulated in finite annular regions. For such approximation problems, we can derive uniform {\it a-priori} estimates that are independent of both the inner and outer radii of the annuli considered in the spherically symmetric Lagrangian coordinates. The entropy inequality is recovered after taking the limit of the outer radius to infinity by using Mazur's lemma and the convexity of the entropy function, which is required for the limit of the inner radius tending to zero. Then the global weak solutions of the original problem are attained via careful compactness arguments applied to the approximate solutions in the Eulerian coordinates.