Asymptotic Behaviour of Solutions of the Stationary Navier‐Stokes Equations
Asymptotic Behaviour of Solutions of the Stationary Navier‐Stokes Equations
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DOI：
10.1112/jlms/s1-44.1.340
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发表时间：
1969
影响因子：
1.2
通讯作者：
R. Dyer;D. Edmunds
中科院分区：
文献类型：
--
作者：
R. Dyer;D. Edmunds
1.在最近的一篇论文[1]中，作者表明，粗略地说，如果u是开连通集Q c R3中定常Navier-Stokes方程的解，并且如果u在fi的某个点处有无限阶的零点，则u在Q中恒为零。这似乎是适当的补充这一结果与一个有关的行为在无穷远的解决方案，在本文中，我们在这个方向上取得了一些进展。当然，关于定常Navier-Stokes方程的外问题，有相当多的文献：我们特别提到Finn [2]的定理：如果u是有界集S的外解，且u在S的边界上等于零，且如果u（x）= ofl*！“1）as\x\-> oo，则u相同地消失。这里我们证明了如果u是S的外解且具有有界的加速度梯度且u（x）= O（exp（-exp（a|.,\-| 3))J为| x |*· oo，对所有a> 0，则u恒为零;与Finn的结果相比，这个结果的相对弱点显然反映了这样一个事实：它仅仅是作为u在无穷远处的局部行为的结果而获得的，证明的方法是使用一个反演，它将无穷大邻域中的问题转化为无穷大邻域中的一个问题。穿孔球，然后使用一个定理的解决方案的微分不等式接近最近得到的Agmon [3]。由于这个定理可能在偏微分方程的一般领域中有一些意义，我们在这里提到它：设v是不等式的C2类解
1. In a recent paper [1] the authors have shown that, roughly speaking, if u is a solution of the stationary Navier-Stokes equations in an open connected set Q c R3, and if u has a zero of infinite order at some point of fi, then u is identically zero in Q. It seems appropriate to supplement this result with one concerning the behaviour at infinity of solutions, and in this paper we make some progress in this direction. There is, of course, a very considerable amount of literature on the exterior problem for the stationary Navier-Stokes equations: we mention in particular the theorem of Finn [2] that if u is a solution in the exterior of a bounded set S, with u equal to zero on the boundary of S, and if u (x)= ofl*!" 1) as\x\-> oo, then u vanishes identically. Here we prove that if u is a solution in the exterior of S with bounded acceleration gradients and u (x)= O (exp (—exp (a|.,\-| 3)) J as\x\-*• oo, for all a> 0, then u is identically zero; the relative weakness of this result in comparison with that of Finn apparently reflects the fact that it is obtained as a consequence of the local behaviour at infinity of u only, without the knowledge of zero boundary data which Finn has.The method of proof is to use an inversion which takes a problem in a neighbourhood of infinity into one in a punctured ball, and then to use a theorem on the solutions of differential inequalities close to those recently obtained by Agmon [3]. Since this theorem is perhaps of some interest in the general field of partial differential equations, we mention it here: Let v be a solution of class C2 of the inequality