论文标题
在Ladyzhenskaya-Serrin上，足以适用于Navier-Stokes方程的常规解决方案。周期性案例
On Ladyzhenskaya-Serrin condition sufficient for regular solutions to the Navier-Stokes equations. Periodic case
论文作者
论文摘要
我们考虑具有周期性边界条件的有界域中的Navier-Stokes方程。令$ v = v（x，t）$为流体的速度。本文的目的是证明任何$ \ | v（t）\ | _ {h^1} \ le c $用于\ at \ mathbb {r} _+$中的任何$ t \，其中$ c $取决于数据。证明分为两个步骤。在第一步中，考虑了带有特殊版本的对流术语的Lamé系统。该系统具有两个粘度。假设第二个粘度（散装粘度）足够大，我们能够证明存在该系统的全球常规解决方案。证明分为两个步骤。首先证明了$ t $与散装粘度成正比的间隔$（0，t）$的长期存在。拥有大粘度大，我们能够证明时间$ t $的数据足够小。然后，通过小数据参数，全球存在随之而来。在本文中，我们仅限于获得适当的估计。为了证明存在，我们应该使用连续的近似方法和持续论点。令$ v $成为解决方案。在第二步中，我们考虑了$ u = v-v $的问题。假设$ \ | u \ | _ {h^1} $ at $ t = 0 $足够小，我们表明$ \ | u（t）\ | _ {h^1} $对于任何$ t \ in \ mathbb {r} _+$也足够小。 $ v $和$ u $ in $ h^1 $的估计意味着$ \ | v（t）\ | _ {h^1} $ in \ in \ mathbb {r} _+$的任何$ t \。
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 $t\in\mathbb{R}_+$.
