三维不可压Leray-α-MHD方程的正则性准则——基于Triebel-Lizorkin空间的研究

Regularity Criteria for the Three-Dimensional Incompressible Leray-α-MHD Equations—A Study Based on Triebel-Lizorkin Spaces

摘要: 本文围绕三维不可压Leray-α-MHD方程的正则性准则展开研究。基于变量分解思想，借助Littlewood-Paley分解理论将$\nabla b$ 分解为三个子模块开展分析，再利用Bernstein不等式和Gagliardo-Nirenberg不等式的组合估计方法，推导得到该方程在Triebel-Lizorkin空间内的正则性判定条件。研究证实，Leray-α-MHD方程的解$\left(u,v,b\right)$ 当$\nabla b\in L^{\frac{4q}{3q-3}}\left(0,T;{\dot{F}}_{q,\frac{4q}{q+3}}^0\left({ℝ}^3\right)\right)$ 时，其光滑性可突破给定时间区间的限制实现延拓。

Abstract: This paper conducts a research on the regularity criterion for the three-dimensional incompressible Leray-α-MHD equations. Based on the idea of variable decomposition, we decompose $\nabla b$ into three sub-modules for analysis by means of the Littlewood-Paley decomposition theory. Then, we employ the combined estimation method of the Bernstein inequality and the Gagliardo-Nirenberg inequality to derive the regularity judgment condition for the equation in the Triebel-Lizorkin space. It is verified that when the solution $\left(u,v,b\right)$ of the Leray-α-MHD equation satisfies $\nabla b\in L^{\frac{4q}{3q-3}}\left(0,T;{\dot{F}}_{q,\frac{4q}{q+3}}^0\left({ℝ}^3\right)\right)$ , its smoothness can be extended beyond the limitation of the given time interval.