AI 中文总结
研究三维抛物型系统作为不可压缩Navier-Stokes方程的近似，结合多种耗散机制。证明任意$L^2$初始数据弱解全局存在，小$H^2$初始数据全局强解存在唯一，$L^1\cap H^2$初始数据有最优衰减估计，且先验估计对惩罚参数一致。
AI 中文摘要
我们研究了一个三维抛物型系统，它是不可压缩Navier-Stokes方程的超粘性和惩罚近似。该模型结合了三种互补的耗散机制：经典粘性扩散、双调和（超粘性）正则化和散度惩罚。此外，在非线性对流项中加入了Temam型修正，以补偿惩罚过程产生的弱压缩效应。我们证明了对于属于$L^2(\mathbb{R}^3)$的任意初始数据，弱解的全局存在性。对于$H^2(\mathbb{R}^3)$中足够小的初始数据，我们建立了全局强解的存在性和唯一性。此外，对于$L^1(\mathbb{R}^3)\cap H^2(\mathbb{R}^3)$中的初始数据，我们导出了最优的大时间衰减估计，表明解具有与经典热方程相同的渐近衰减率。我们分析的一个关键特征是，所有得到的先验估计对于正惩罚参数$\varepsilon$是一致的。这些一致的界为不可压缩流的惩罚近似研究提供了一个稳定而严格的分析基础。
英文摘要
We investigate a three-dimensional parabolic system that arises as a hyperviscous and penalized approximation of the incompressible Navier--Stokes equations. The model combines three complementary dissipative mechanisms: the classical viscous diffusion, a biharmonic (hyperviscous) regularization, and a divergence penalization. In addition, a Temam-type correction is incorporated into the nonlinear convection term to compensate for the weak compressibility effects generated by the penalization procedure. We prove the global existence of weak solutions for arbitrary initial data belonging to $L^2(\mathbb{R}^3)$. For sufficiently small initial data in $H^2(\mathbb{R}^3)$, we establish the existence and uniqueness of global strong solutions. Furthermore, for initial data in $L^1(\mathbb{R}^3)\cap H^2(\mathbb{R}^3)$, we derive optimal large-time decay estimates, showing that the solutions exhibit the same asymptotic decay rates as those of the classical heat equation. A key feature of our analysis is that all the obtained a priori estimates are uniform with respect to the positive penalization parameter $\varepsilon$. These uniform bounds provide a stable and rigorous analytical foundation for the study of the penalized approximation of incompressible flows.
