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

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 $\eta {\mathrm{\mid }}_{t=0}$ is sufficiently small in $H^2$-norm. We assume existence of potentials $\phi$ and $\psi$ such that $v=\mathrm{\nabla }\phi +\mathrm{r}\mathrm{o}\mathrm{t}\psi$. Next we assume that $\mathrm{\nabla }\phi {\mathrm{\mid }}_{t=0}$ 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 $v\in L_{\mathrm{\infty }}(0,T;H^2(\mathrm{\Omega }))\cap L_2(0,T;H^3(\mathrm{\Omega }))$, $v_{,t}\in L_\infty(0,T;H^1(\Omega))\cap L_2(0,T;H^2(\Omega))$,wheretheexistencetime$T$isproportionalto$\nu$\mathrm{.}Nextfor$T$sufficientlylargeweobtainthat$v(T)$ is correspondingly small so global existence is proved using the methods appropriate for problems with small data.