Existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force

TL;DR

This paper investigates the existence, stability, and long-term behavior of weak solutions to the three-dimensional compressible Navier-Stokes equations with potential force, focusing on how solutions depend on initial data and their convergence over time.

Contribution

It provides new insights into the long-time behavior, stability, and initial data dependence of weak solutions for the 3D compressible Navier-Stokes equations with potential force.

Findings

Weak solutions exhibit specific smoothing rates near t=0 depending on initial velocity regularity.

Weak solutions converge in various norms over long times.

Weak solutions depend continuously on initial data and steady states.

Abstract

We address the global-in-time existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We show the details of the $\alpha$-dependence of different smoothing rates for weak solutions near $t=0$ under the assumption on the initial velocity $u_0$ that $u_0\in H^\alpha$ for $\alpha\in(\frac{1}{2},1]$ and obtain long time convergence of weak solutions in various norms. We then make use of the Lagrangean framework in comparing the instantaneous states of corresponding fluid particles in two different solutions. The present work provides qualitative results on the long time behaviour of weak solutions and how the weak solutions depend continuously on initial data and steady states.