Existence of regular solutions to the stationary Navier-Stokes equations
Existence of regular solutions to the stationary Navier-Stokes equations
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DOI：
10.1007/bf01444513
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发表时间：
1995-05
影响因子：
1.4
通讯作者：
J. Frehse;M. Růžička
中科院分区：
文献类型：
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作者：
J. Frehse;M. Růžička
-au+u 9Vu+V‘p=f在q(1.1)divu=0中具有周期边界条件，其中~=(0，L)N，5_<；N<；10.在以前的文献[2]中，[3]我们证明了某些弱解的存在性，对于这些弱解，下面给出fqp1dx<；c(F)sup+ix-x0lN-2=x0EI2(1.2)sup fku.“(x-xo))2dx<；c(F)xo~ix”xN=hold.让我们在这里强调，(1.2)的证明基本上使用了对流项u9的结构。一旦建立了(1.2)，人们就可以在某些Morrey空间中分别获得关于速度u和压力p的额外估计，不幸的是，其中一个人失去了任意的小重量。本文导出了仅依赖于外力f的L~(I2)范数的C~中速度u和C~中压力p的先验估计，这些估计直接通过一个简单的Bootstrap引理隐含正则性。在证明中，我们主要使用(1.2)2和证明中使用的表示公式，以表明对于所有e>；0，都有Po>；0使得
-Au+ u 9 Vu+ V'p= f in Q(1.1) divu= 0 with periodic boundary conditions, where~=(0, L) N, 5 __< N< 10. In previous papers [2],[3] we showed the existence of certain weak solutions for which the following" optimal" estimates fqp 1 dx< c (f) sup+ Ix-x0l N-2= x0EI2 (1.2) sup fku."(x-xo)) 2 dx< c (f) xo~~ Ix" x01 N= hold. Let us emphasize here that the proof of (1.2) essentially uses the structure of the convective term u 9 gru.Once having established (1.2) one can obtain additional estimates in certain Morrey spaces for the velocity u and the pressure p separately, in which unfortunately one loses an arbitrary small weight. In this paper we derive apriori estimates for the velocity u in C~ and for the pressure p in C~ which depend only on the L~(I2)-norm of the outer force f. These estimates immediately imply regularity via a simple bootstrap argument. In the proof we essentially use (1.2) 2 and a representation formula used in the proof in order to show that for all e> 0 there is a Po> 0 such that