On Ladyzhenskaya-Serrin condition sufficient for regular solutions to the Navier-Stokes equations. Periodic case

We consider the Navier-Stokes equations in a bounded domain with periodic boundary conditions. Let $V=V(x,t)$ be the velocity of the fluid. The aim of this paper is to prove the bound $\|V(t)\|_{H^1}\le c$ for any $t\in\mathbb{R}_+$, where $c$ depends on data. The proof is divided into two steps. In the first step the Lamé system with a special version of the convective term is considered. The system has two viscosities. Assuming that the second viscosity (the bulk one) is sufficiently large we are able to prove the existence of global regular solutions to this system. The proof is divided into two steps. First the long time existence in interval $(0,T)$ is proved, where $T$ is proportional to the bulk viscosity. Having the bulk viscosity large we are able to show that data at time $T$ are sufficiently small. Then by the small data arguments a global existence follows. In this paper we are restricted to derive appropriate estimates only. To prove the existence we should use the method of successive approximations and the continuation argument. Let $v$ be a solution to it. In the second step we consider a problem for $u=v-V$. Assuming that $\|u\|_{H^1}$ at $t=0$ is sufficiently small we show that $\|u(t)\|_{H^1}$ is also sufficiently small for any $t\in\mathbb{R}_+$. Estimates for $v$ and $u$ in $H^1$ imply estimate for $\|V(t)\|_{H^1}$ for any $t\in\mathbb{R}_+$.

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