Global Solutions of the Navier-Stokes Equations for Isentropic Flow with Large External Potential Force

TL;DR

This paper proves the global existence of weak solutions to the 3D compressible Navier-Stokes equations with large external forces, providing insights into regularity and long-term behavior.

Contribution

It establishes the global existence of weak solutions for isentropic flow with large external potential forces, even when the adiabatic exponent is one, without smallness assumptions on external forces.

Findings

Global weak solutions exist for the equations considered.

Partial regularity and large-time behavior are characterized.

Solutions are valid for large external forces when γ=1.

Abstract

We prove the global-in-time existence of weak solutions to the Navier-Stokes equations of compressible isentropic flow in three space dimensions with adiabatic exponent $\gamma\ge1$. Initial data and solutions are small in $L^2$ around a non-constant steady state with densities being positive and essentially bounded. No smallness assumption is imposed on the external forces when $\gamma=1$. A great deal of information about partial regularity and large-time behavior is obtained.