带非线性滤波方程的新型Navier-Stokes-α模型的数学研究——正则性理论与精细时间渐近性
Mathematical study of a new Navier-Stokes-alpha model with nonlinear filter equation - Regularity theory and refined time asymptotics
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中文总结 AI 辅助
本文研究带非线性滤波方程的新型Navier-Stokes-α模型，建立了全局吸引子的一致高阶正则性，并证明了吸引子上完整轨迹的精确确定模性质，即若两条轨迹在足够大的有限维Stokes模空间上的投影处处重合，则轨迹完全相同。
中文摘要 AI 辅助
本文致力于一个涉及非线性滤波方程的新型Navier-Stokes-α模型的数学分析。该模型由一个双重非线性抛物-椭圆耦合系统控制。在我们先前的工作中（M. F. Cortez和O. Jarrín，《带非线性滤波方程的新型Navier-Stokes-α模型的数学研究——第一部分》，J. Math. Fluid Mech. 28, no. 12 (2026)），我们建立了弱Leray型解的全局适定性和全局吸引子的存在性。在本工作中，在关于A(·)的自然假设下，我们研究了该模型的若干附加性质，特别强调高阶正则性及其对长时间动力学的影响。正则性分析构成一个核心且微妙的问题，因为椭圆滤波器的非线性结构及其与演化方程的耦合，排除了直接应用线性滤波Navier-Stokes-α模型可用的标准正则性论证的可能性。克服这一困难需要对非线性椭圆滤波方程建立新的高阶估计，这些估计可能也具有独立的意义。在所得结果中，有两个构成了本文的主要贡献。首先，我们建立了全局吸引子的一致高阶正则性。该结果随后被用作证明吸引子上完整轨迹的精确确定模性质的基本要素：如果两条完整轨迹在足够大的有限维Stokes模空间上的投影在每个时刻都重合，那么这两条轨迹完全一致。
英文摘要
This article is devoted to the mathematical analysis of a new Navier--Stokes-$α$ model involving a nonlinear filter equation. The resulting model is governed by a doubly nonlinear parabolic--elliptic coupled system. In our previous work [M.~F.~Cortez and O.~Jarrín, \emph{Mathematical study of a new Navier--Stokes-alpha model with nonlinear filter equation -- Part I}, J. Math. Fluid Mech. 28, no.~12 (2026)], we established the global well-posedness of weak Leray-type solutions and the existence of a global attractor. In the present work, under natural assumptions on $A(\cdot)$, we investigate several additional properties of this model, with particular emphasis on higher-order regularity and its consequences for the long-time dynamics. The regularity analysis constitutes a central and delicate issue, since the nonlinear structure of the elliptic filter and its coupling with the evolution equation preclude a direct application of the standard regularity arguments available for linearly filtered Navier--Stokes-$α$ models. Overcoming this difficulty requires new higher-order estimates for the nonlinear elliptic filter equation, which may also be of independent interest. Among the results obtained, two constitute the main contributions of the article. First, we establish uniform higher-order regularity for the global attractor. This result is then used as a fundamental ingredient in proving an exact determining-modes property for complete trajectories on the attractor: if the projections of two complete trajectories onto a sufficiently large finite-dimensional space of Stokes modes coincide at every time, then the two trajectories coincide identically.
