发表机构
School of Mathematics and Statistics, Nanjing University of Science and Technology（南京理工大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对二维Navier--Stokes方程水平黏性情形，证明Lions条件与垂直方向Onsager临界正则性蕴含水平耗散正则性，并进一步得到各向同性临界空间、能量等式与L^2连续性。
AI 中文摘要
我们考虑环面$\mathbb T^2$或$\mathbb R^2$上具有水平黏性的二维Navier--Stokes方程的分布意义弱解$u\in L^\infty_tL^2_x$，不假设$\partial_1u\in L^2_tL^2_x$。我们证明Lions可积性条件$u\in L^4_tL^4_x$连同非耗散方向上的Onsager临界正则性$u\in L^3_tB^{1/3,v}_{3,\infty}$（其中Besov正则性仅施加于垂直方向）蕴含$\partial_1u\in L^2_tL^2_x$，并给出水平耗散的定量界。因此耗散正则性是方程的结果，不必作为解定义的一部分。此外，若$u\in L^3_tB^{1/3,v}_{3,c_0}$，则$u$属于各向同性临界Onsager空间$L^3_tB^{1/3}_{3,c_0}$，满足能量等式，且在$L^2$中连续。这里$c_0$表示相应的二进Besov序列在高频趋于零。证明将水平能量通量吸收进耗散与垂直能量通量的一维交换子估计相结合。
英文摘要
We consider distributional weak solutions $u\in L^\infty_tL^2_x$ of the two-dimensional Navier--Stokes equations with horizontal viscosity on $\mathbb T^2$ or $\mathbb R^2$, without assuming $\partial_1u\in L^2_tL^2_x$. We show that the Lions integrability condition $u\in L^4_tL^4_x$ together with Onsager-critical regularity in the nondissipative direction, $u\in L^3_tB^{1/3,v}_{3,\infty}$ with the Besov regularity imposed only in the vertical direction, implies $\partial_1u\in L^2_tL^2_x$, with a quantitative bound on the horizontal dissipation. Thus the dissipative regularity is a consequence of the equation and need not be part of the definition of the solution. If moreover $u\in L^3_tB^{1/3,v}_{3,c_0}$, then $u$ belongs to the isotropic critical Onsager space $L^3_tB^{1/3}_{3,c_0}$, satisfies the energy equality, and is continuous in $L^2$. Here $c_0$ indicates that the corresponding dyadic Besov sequence tends to zero at high frequencies. The proof combines absorption of the horizontal energy flux into the dissipation with a one-dimensional commutator estimate for the vertical energy flux.
