不可压缩Navier-Stokes方程系统的空间批判
Espaces critiques pour le système des equations de Navier-Stokes incompressibles
摘要：
在这项工作中，我们展示了抽象的功能空间E上的条件，确保存在一个整体的温和的解决方案的小数据在E或存在的局部温和的解决方案，在没有大小限制的一类半线性抛物型方程，其中包含不可压缩的Navier-Stokes系统作为一个基本的例子。我们还给出了一个抽象的判据，对所获得的解决方案的正则性。这些条件，在Littlewood-Paley估计产品的频谱本地化元素的$E$，是简单的检查在所有已知的情况下：勒贝格，洛伦茨，Besov，Morrey.空间.这些条件也适用于非不变空间E，我们给出了一些2-微局部空间的情况下的全部细节。以下评论没有出现在第一版上：这篇文章写于1998-99年左右，从未发表过，因为当时Koch和Tataru在$BMO^{-1}$中公布了他们关于Navier-斯托克斯方程的适定性的结果。我们相信，虽然这里的一些结果和反例是独立的利益，我们使他们电子。
英文摘要：
In this work, we exhibit abstract conditions on a functional space E who insure the existence of a global mild solution for small data in E or the existence of a local mild solution in absence of size constraints for a class of semi-linear parabolic equations, which contains the incompressible Navier-Stokes system as a fundamental example. We also give an abstract criterion toward regularity of the obtained solutions. These conditions, given in terms of Littlewood-Paley estimates for products of spectrally localized elements of $E$, are simple to check in all known cases: Lebesgue, Lorents, Besov, Morrey... spaces. These conditions also apply to non-invariant spaces E and we give full details in the case of some 2-microlocal spaces. The following comments did not show on the first version: This article was written around 1998-99 and never published, because at that time, Koch and Tataru announced their result on well-posedness of Navier-stokes equations with initial data in $BMO^{-1}$. We believe though that some results and counterexamples here are of independent interest and we make them available electronically.
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