标题： 不可压缩Navier-Stokes方程的统一频域重构与边界适应 显示英文标题

标题： Unified Frequency-Domain Reconstruction and Boundary Adaptation for Incompressible Navier-Stokes Equations

摘要： 三维不可压缩纳维-斯托克斯方程的全局正则性问题仍未解决，部分原因在于弱解、温和解和强解的表述采用了不兼容的功能设置。 本研究引入了一个频域框架，将这些表述整合到一个统一的构造方案中。 从勒雷-霍普夫数据出发，一种结合平滑、索博列夫延拓和勒雷投影器的尺度依赖正则化算子生成光滑且散度为零的近似值。 然后定义了两个有界的傅里叶乘子：一个插值算子，用于混合低频弱分量和高频强分量，另一个平滑算子，产生均匀抛物增益。 利特尔伍德-帕莱分析、改进的卡达尔通-齐格蒙德和施劳德估计，以及显式伽辽金方案提供了定量的Hs先验界，并且寿命由第一个斯托克斯特征值控制。 通过奥宾-利翁斯引理和控制收敛定理证明，当正则化参数消失时，近似值收敛到一个满足同时弱能量恒等式、福田-加藤温和表述以及Hs中的强表述（s > 3/2）的单一速度场。 该构造提供了一个统一的局部解，并提供了后续能量-涡度自举和全局延续论证所需的算子界，阐明了频率局部化与流体动力学中非线性估计之间的相互作用。 显示英文摘要

摘要： Global regularity for the three-dimensional incompressible Navier-Stokes equations remains unresolved partly because weak, mild, and strong formulations employ incompatible functional settings. The present study introduces a frequency-domain framework that reconciles these formulations within a single constructive scheme. Starting from Leray-Hopf data, a scale-dependent regularization operator combining mollification, Sobolev extension, and the Leray projector produces smooth, divergence-free approximations. Two bounded Fourier multipliers are then defined: an interpolation operator that blends low-frequency weak and high-frequency strong components, and a smoothing operator that yields uniform parabolic gain. Littlewood-Paley analysis, refined Calderon-Zygmund and Schauder estimates, and an explicit Galerkin scheme provide quantitative Hs a priori bounds and a lifespan controlled by the first Stokes eigenvalue. Compactness via the Aubin-Lions lemma and dominated convergence demonstrates that, as the regularization parameter vanishes, the approximations converge to a single velocity field satisfying simultaneously the weak energy identity, the Fujita-Kato mild formulation, and the strong formulation in Hs with s > 3/2. The construction furnishes a unified local solution and supplies operator bounds required for subsequent energy-vorticity bootstraps and global continuation arguments, clarifying the interplay between frequency localization and nonlinear estimates in fluid dynamics.