Global regular periodic solutions to equations of weakly compressible barotropic fluid motions

TL;DR

This paper proves the long-time existence of solutions for weakly compressible barotropic fluid equations with periodic boundary conditions, showing solutions remain regular over time and global existence follows for small data.

Contribution

It establishes the existence of global regular solutions for the compressible Navier-Stokes equations under specific smallness and viscosity conditions, extending previous local results.

Findings

Solutions exist for a time proportional to viscosity.

Solutions remain in high regularity spaces over time.

Global existence is achieved for sufficiently small initial data.

Abstract

We consider barotropic motions described by the compressible Navier-Stokes equations in a box with periodic boundary conditions. We are looking for density $\varrho$ in the form $\varrho=a+\eta$, where $a$ is a constant and $\eta|_{t=0}$ is sufficiently small in $H^2$-norm. We assume existence of potentials $\varphi$ and $\psi$ such that $v=\nabla\varphi+\mathrm{rot}\psi$. Next we assume that $\nabla\varphi|_{t=0}$ is sufficiently small in $H^2$-norm too. Finally, we assume that the second viscosity coefficient $\nu$ is sufficiently large. Then we prove long time existence of solutions such that $v\in L_\infty(0,T;H^2(\Omega))\cap L_2(0,T;H^3(\Omega))$, $v_{,t}\in L_\infty(0,T;H^1(\Omega))\cap L_2(0,T;H^2(\Omega))$, where the existence time $T$ is proportional to $\nu$. Next for $T$ sufficiently large we obtain that $v(T)$ is correspondingly small so global existence is proved using the methods appropriate for problems with small data.