  We find a global a priori estimate for solutions to the Navier-Stokes
equations with periodic boundary conditions guaranteeing in view of the Serrin
type condition the existence of global regular solutions. We derive the
following estimate $$ \lVert V(t) \rVert_{H^1(\Omega)}\leq c, \qquad (1) $$
where $V$ is the velocity of the fluid. The estimate (1) is proved in two
steps. First we derive a global estimate guaranteeing the existence of global
regular solutions to weakly compressible Navier-Stokes equations with large
second viscosity, density close to a constant and gradient part of velocity
small. Next we show that solutions to the Navier-Stokes equations remain close
to solutions to the weakly compressible Navier-Stokes equations if the
corresponding initial data and external forces are sufficiently close.

Analysis of PDEs