三维各向异性Navier-Stokes方程在临界空间中解的稳定性

中文摘要 AI 辅助

本文研究了初值位于临界空间$\dot{B}^{0,\frac12}$中的三维不可压缩各向异性Navier-Stokes方程$(ANS)$解的连续依赖性。我们证明了在该设定下数据到解的映射是连续的。由于问题的临界正则性，经典的Bona-Smith论证\cite{BoSm}（另见抽象结果\cite{ABITZ}）似乎不适用，因为一个关键要素（较低正则性下的Lipschitz连续性）似乎不成立。因此，我们在Besov框架内发展了一种新的直接傅里叶加权方法。

英文摘要

In this paper, we study the continuous dependence of solutions to the three-dimensional incompressible anisotropic Navier-Stokes equations $(ANS)$ with initial data in the critical space $\dot{B}^{0,\frac12}$. We prove that the data-to-solution map is continuous in this setting. Due to the critical regularity of the problem, the classical Bona-Smith argument \cite{BoSm} (see also the abstract result \cite{ABITZ}) does not appear to apply because a key ingredient (the Lipschitz continuity at lower regularity) does not appear to hold. As a consequence, we develop a new direct Fourier-weighted method within the Besov framework.

发表机构

• Institut Universitaire de France（法兰西学院）
• Department of Mathematics, New York University Abu Dhabi（纽约大学阿布扎比分校数学系）
• School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）

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