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

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 $\varrho=a+η$, where $a$ is a constant and $η|_{t=0}$ is sufficiently small in $H^2$-norm. We assume existence of potentials $φ$ and $ψ$ such that $v=\nablaφ+\mathrm{rot}ψ$. Next we assume that $\nablaφ|_{t=0}$ is sufficiently small in $H^2$-norm too. Finally, we assume that the second viscosity coefficient $ν$ is sufficiently large. Then we prove long time existence of solutions such that $v\in L_\infty(0,T;H^2(Ω))\cap L_2(0,T;H^3(Ω))$, $v_{,t}\in L_\infty(0,T;H^1(Ω))\cap L_2(0,T;H^2(Ω))$, where the existence time $T$ is proportional to $ν$. 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.