The exterior nonstationary problem for the Navier-Stokes equations
The exterior nonstationary problem for the Navier-Stokes equations
复制标题
纳维-斯托克斯方程的外部非平稳问题
DOI：
10.1007/bf02392212
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发表时间：
1972
期刊：
影响因子：
3.7
通讯作者：
J. G. Heywood
中科院分区：
文献类型：
--
作者：
J. G. Heywood
虽然Leray[1]已经证明了Navier-Stokes方程的外平稳问题在非常一般的情况下有解，但即使在小数据的情况下，该问题的Leray解是否唯一，或者它是否可以形成为t~o的非定常解的极限，仍然是未知的。在这篇文章中，我们证明了对于一类特定的给定边值，恰好存在一个定常解，作为物理上合理的非定常解的HMIT，我们的方法是基于初边值问题的一个整体存在定理，该定理是在允许无穷远的含时边值和含时速度的假设下证明的。这一定理保证了当存在足够好且满足稳定性条件的近似解时，初边值问题的唯一可解性。这一存在定理也使我们能够给出保证任意三维区域上定义的I~ille-Stokes方程非定常解的稳定性的简单条件。在粘性不可压缩流动理论中，流体运动主要由N-S方程控制。L方程的外定常问题是在有界曲面外的区域内找到与时间无关的速度函数和压力函数，它们共同求解方程，使得速度函数在表面上取给定值，在无穷远处趋于规定的极限值。当然，在自然界中，定常流动只是作为非定常流动的极限。外部静止问题的解可能是流体流动的模型，可以通过正确选择规定的数据进行以下理想化的实验来获得。一个物体沉浸在一种占据整个三维空间的流体中
Although the exterior stationary problem for the Navier-Stokes equations has been proved by Leray [1] to possess a solution under very general circumstances, it is unknown even in the case of small data whether Leray's solution of the problem is unique or whether it may be formed as the limit of a nonstationary solution as t~ oo. In this paper we prove that for a particular class of prescribed boundary values there is exactly one stationary solution attainable as the hmit, starting from rest, of a physically reasonable nonstationary solutiom Our method is based on a global existence theorem for the initial boundary value problem which we prove under hypotheses that allow time dependent boundary values and a time dependent velocity at infinity. This theorem assures the unique solvability of the initial boundary value problem whenever there is an approximate solution which is sufficiently good and satisfies a stability condition. This existence theorem has also enabled us to state simple conditions sufficient to ensure the stability of nonstationary solutions of the I~ avier-Stokes equations defined in arbitrary three-dimensional regions.The Navier-Stokes equations govern fluid motion in the theory of viscous incompressible flow. The exterior stationary problem for the l~ avier-Stokes equations consists of finding, in the region exterior to a dosed bounded surface, time independent velocity and pressure functions which together solve the equations and are such that the velocity function assumes given values on the surface and tends to a prescribed limit at infinity. Of course, stationary flow occurs in nature only as the limit of nonstationary flow. Presumably solutions of the exterior stationary problem model fluid flows which may be obtained by performing the following ideahzed experiment with the right choice of prescribed data. An object is immersed in a fluid which occupies all three-dimensional