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We consider barotropic motions described by the compressible Navier-Stokesequations in a box with periodic boundary conditions. We are looking fordensity $\varrho$ 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 ofpotentials $\varphi$ and $\psi$ such that $v=\nabla\varphi+\mathrm{rot}\psi$.Next we assume that $\nabla\varphi|_{t=0}$ is sufficiently small in $H^2$-normtoo. Finally, we assume that the second viscosity coefficient $\nu$ issufficiently large. Then we prove long time existence of solutions such that$v\in L_\infty(0,T;H^2(\Omega))\cap L_2(0,T;H^3(\Omega))$, $v_{,t}\inL_\infty(0,T;H^1(\Omega))\cap L_2(0,T;H^2(\Omega))$, 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 methodsappropriate for problems with small data.

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