二维超粘Couette流方程的Sobolev稳定性
The Sobolev Stability of 2D Hyperviscosity Equations for Couette Flow
摘要: 近年来，等离子体场方面的突破激发了流体力学中非线性无粘阻尼的研究，并且已有证明针对Gevrey类扰动的Couette流周围的二维欧拉方程的非线性无粘阻尼是成立的。在此基础上，本文研究了分数阶二维超粘性方程在Sobolev空间中Couette流的渐近稳定性和增强耗散性，通过线性化的方式得到了该方程具有无粘阻尼和增强耗散，并且通过构造适合的权重进行Bootstrap论证，发现了如果Couette流有充分小的扰动，由于混合增强耗散效应，解在时间充分大时收敛。因此，得出结论：具有初值的二维超粘性方程的稳定性阈值不比某定值差。
Abstract: In recent years, breakthroughs in plasma fields have stimulated the study of nonlinear inviscous damping in fluid mechanics, and it has been demonstrated that the nonlinear inviscous damping of the two-dimensional Euler equation around the Couette flow of Gevrey-class perturbations is true. In this paper, on this basis, the asymptotic stability and enhanced dissipation of the fractional-order two-dimensional hyperviscosity equation for Couette flow in the Sobolev space are studied, and the equation has inviscid damping and enhanced dissipation by linearization, and the Bootstrap argument by constructing suitable weights shows that if the Couette flow has a sufficiently small perturbation, the solution converges when the time is sufficiently large due to the mixed-enhanced dissipation effect. Therefore, it is concluded that the stability threshold of a two-dimensional hyperviscosity equation with an initial value is not worse than that of a certain fixed value.
文章引用：金自地, 刘晓风. 二维超粘Couette流方程的Sobolev稳定性[J]. 理论数学, 2024, 14(7): 181-194. https://doi.org/10.12677/pm.2024.147285

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