On uniqueness questions in the theory of viscous flow
On uniqueness questions in the theory of viscous flow
复制标题
论粘性流动理论中的唯一性问题
DOI：
10.1007/bf02392043
复制
发表时间：
1976
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
本文通过反例证明，对于某些空间区域，粘性不可压缩流体的运动并不唯一地由传统的初始和边界条件（即由所施加的外力和流体在初始时刻、空间区域边界和空间无穷远处的速度值）决定。在一个积极的方向，我们证明了唯一性的初始边值问题在某些类别的空间域，并在适当的辅助条件下，这个问题在其他类别的domMns唯一。考虑到这里所要考虑的唯一性问题，我们将表明，定常流问题与非定常流问题的情形是相同的，线性Stokes方程与非线性Navier-Stokes方程的情形也是相同的。在某些方面，我们的结果是在方差~% h在以前的作品中给出的主题，并在其他一些方面，我们的结果可能出现在第一次是不是新的。其中最重要的文件理论流体力学是一些调查的存在性和唯一性理论的边界值问题的粘性流动各类广义解。相当值得注意的是，这些以前研究的广义解类的唯一性证明并没有利用空间域的任何性质，因此它们的唯一性定理在许多情况下都是针对任意空间域给出的。这是在Ladyzhenskaya的著名作品[21]，[31]普罗迪，[33] Serrin，也在本作者的论文[14，16，17]轻松。这些文件的唯一性定理是误导，但是，因为类的广义解决方案，他们适用于已被定义在这样一种方式，以排除成员，在某些领域，一些经典的和物理上重要的解决方案。
In this paper it is shown, by means of counter examples, that for some spatial domains tile motion of a viscous incompressible fluid is not uniquely determined by the traditional initial and boundary conditions (ie, by the applied external forces and by the values of the fluid velocity at an initial instant of time, at the boundary of the spatial domain, and at spatial infinity). In a positive direction, we prove uniqueness for the initial boundary value problem in some classes of spatial domains, and uniqueness for this problem in other classes of domMns under appropriate auxiliary conditions. I~ egarding the uniqueness questions to be considered here, it will be shown that the situation is much the same for the problems of steady flow as for those of nonstationary flow, and much the same for the linear Stokes equations as for the nonlinear Navier-Stokes equations. In some respects our results are at variance~% h those given in previous works on the subject, and in some other respects our results may appear at first to be not new. Among the most important papers on theoretical hydrodynamics are some investigations of the existence and uniqueness theory for the boundary value problems of viscous flow within various classes of generalized solutions. Rather remarkably, the uniqueness proofs for these previously studied generalized solution classes do not make use of any properties of the spatial domain, and so the uniqueness theorems for them have in many eases been given for an arbitrary spatial domain. This is the ease in the celebrated works [21] of Ladyzhenskaya,[31] of Prodi, and [33] of Serrin, and also in the present author's papers [14, 16, 17]. The uniqueness theorems of these papers are misleading, however, because the classes of generalized solutions to which they apply have been defined in such a way as to exclude from membership, in some domains, some classical and physically important solutions.