Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$
Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$
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DOI：
10.1016/j.anihpc.2007.05.008
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发表时间：
2006-11
影响因子：
1.9
通讯作者：
J. Chemin;I. Gallagher
中科院分区：
文献类型：
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作者：
J. Chemin;I. Gallagher
在[J. Y.舍明岛Gallagher，On the global wellposedness of the 3-D Navier-Stokes equations with large initial data，Annales Scientifiques de l 'École Normale Supérimonde巴黎，in press]提出了一类三维周期性不可压缩Navier-Stokes方程的初始数据，尽管初始数据的范数可以选择任意大，但可以生成全局光滑解。本文的目的是双重的。首先，我们调整[J]的结构。Y.舍明岛Gallagher，On the global wellposedness of the 3-D Navier-Stokes equations with large initial data，Annales Scientifiques de l 'École Normale Supérimonde巴黎，in press]到整个空间的情况：我们证明了，如果初始数据的某个非线性函数足够小，在Koch-Tataru [H. Koch，D. Tataru，Well-posedness for the Navier-Stokes equations，Advances in Mathematics 157（2001）22-35]型空间，则存在Navier-Stokes方程的整体解。我们提供了一个初始数据满足非线性小条件的例子，但其范数在C−1中任意大。然后在非线性小性假设下证明了一个稳定性结果。更确切地说，我们表明，新的小假设也适用于初始数据的平移和扩张迭代的线性叠加，在[H。Bahouri，J.- Y.舍明岛Gallagher，Refined哈代inequalities，Annali di Scuola Normale di比萨，Classe di Scienze，Serie V 5（2006）375-391]，从而产生了大量不同的例子。
In [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'École Normale Supérieure de Paris, in press] a class of initial data to the three dimensional, periodic, incompressible Navier–Stokes equations was presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction of [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'École Normale Supérieure de Paris, in press] to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch–Tataru [H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Advances in Mathematics 157 (2001) 22–35] type space, then there is a global solution to the Navier–Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large in C−1. Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction in [H. Bahouri, J.-Y. Chemin, I. Gallagher, Refined Hardy inequalities, Annali di Scuola Normale di Pisa, Classe di Scienze, Serie V 5 (2006) 375–391], thus generating a large number of different examples.