The initial value problem for the navier-stokes equations
The initial value problem for the navier-stokes equations
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DOI：
10.1007/bf00282248
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发表时间：
1966
影响因子：
2.5
通讯作者：
M. Shinbrot;S. Kaniel
中科院分区：
文献类型：
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作者：
M. Shinbrot;S. Kaniel
1. LERAY [1]的一个美丽的定理指出，如果u（x，t）是Navier-Stokes方程的弱解，对R3中的所有x，则u实际上是光滑的，除了可能在t的值的集合E上具有以下性质：E的Lebesgue测度为零，E的补是区间并;最后一个这样的区间是半无限的，所以所有弱解都是最终光滑的。这个定理的美丽只是因为要求解存在于所有空间中而不是任意光滑区域中这一事实而受损。作为本文主要定理的应用，我们证明了LERAY定理中的这个缺陷是可以消除的。为此，我们首先证明了Navier-Stokes方程解的存在性定理。概括地说，存在理论的现状是这样的。设D是R a中的有界区域.给定一个初始时刻t0，并给定在L2（D）中具有两个分布导数的初始数据，存在一个时间间隔（t0，t1），其中Navier-Stokes方程具有在t0处采用给定数据的强解。这是KISELEV & LADYZHENSKAIA [2，3]的著名结果。如果在一定意义下，初始数据足够小，tt= oo，并且这个光滑解存在于所有时间。但对于大的初始数据，情况可能并非如此。
1. A beautiful theorem of LERAY'S [1] states that if u (x, t) is a weak solution of the Navier-Stokes equations for all x in R 3, then u is actually smooth except possibly on a set E of values of t having the following properties: the Lebesgue measure of E is zero; the complement of E is a union of intervals; and the last such interval is semi-infinite, so that all weak solutions are ultimately smooth. The beauty of this theorem is only marred by the fact that the solution is required to exist in all space rather than in an arbitrary smooth domain. As an application of the main theorem proved here, we show that this flaw in LERAY'S theorem can be removed.To do this, we first prove an existence theorem for solutions of the Navier-Stokes equations. In broad outline, the present state of existence theory is this. Let D be a bounded domain in R a. Given an initial instant to, and given initial data with two distributional derivatives in L2 (D), there is an interval (to, tl) of time in which the Navier-Stokes equations have a strong solution taking on the given data at to. This is the celebrated result of KISELEV & LADYZHENSKAIA [2, 3]. If, in a certain definite sense, the initial data are small enough, tt= oo, and this smooth solution exists for all time. But this may fail to be the case for large initial data.