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arXiv · 1810.04928

On regular periodic solutions to the Navier-Stokes equations

Wojciech M. Zajaczkowski
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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.

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Wojciech M. Zajaczkowski. 2018-10-11. On regular periodic solutions to the Navier-Stokes equations. https://arxiv.org/abs/1810.04928

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