On regular periodic solutions to the Navier-Stokes equations

TL;DR

This paper establishes a global a priori estimate for solutions to the Navier-Stokes equations with periodic boundary conditions, ensuring global regularity under certain conditions, and connects solutions of compressible and incompressible flows.

Contribution

It provides a new global estimate for Navier-Stokes solutions and links compressible and incompressible flows under specific initial conditions.

Findings

Derived a global $H^1$ estimate for velocity.

Proved existence of global regular solutions under certain conditions.

Showed solutions remain close to compressible flow solutions if initial data are close.

Abstract

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.