一个修正的超粘性Navier-Stokes方程的Sobolev稳定性

The Sobolev Stability of a Modified Hyperviscous Navier-Stokes Equation

摘要: 本文主要考虑$T\times R$ 上的二维修正的超粘性Navier-Stokes方程，通过对方程进行线性化处理，揭示了其无粘阻尼特性以及增强耗散现象。进一步地，借助构造合适的权重函数，并运用Bootstrap论证方法，研究发现，当Couette流受到足够小的扰动时，混合增强耗散效应将显著发挥作用，解在时间$t\gg {\nu }^{\frac{1}{5}}$ 时收敛(其中$\nu$ 表示运动粘度系数)。因此，可以得出结论：具有初值的二维修正的超粘性Navier-Stokes方程的稳定性阈值不比${\nu }^{\frac{1}{2}}$ 差。

Abstract: This paper primarily investigates the two-dimensional modified hyperviscous Navier-Stokes equations on $T\times R$ . By linearizing the equations, we reveal their inviscid damping properties and enhanced dissipation phenomena. Furthermore, through the construction of appropriate weight functions and the application of the Bootstrap argument, we find that when the Couette flow is subjected to sufficiently small perturbations, the enhanced dissipation effect due to mixing becomes significant, and the solution converges in time at a rate of $t\gg {\nu }^{\frac{1}{5}}$ (where $\nu$ denotes the kinematic viscosity coefficient). Therefore, we can conclude that the stability threshold for the two-dimensional modified hyperviscous Navier-Stokes equations with initial values is no worse than that of ${\nu }^{\frac{1}{2}}$ .