On regular periodic solutions to the Navier-Stokes equations

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.