3D anisotropic Navier–Stokes equations in T2×R : stability and large-time behaviour
3D anisotropic Navier–Stokes equations in T2×R : stability and large-time behaviour
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DOI：
10.1088/1361-6544/acd160
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发表时间：
2023-05
期刊：
影响因子：
1.7
通讯作者：
Ruihong Ji;Ling Tian;Jiahong Wu
中科院分区：
文献类型：
--
作者：
Ruihong Ji;Ling Tian;Jiahong Wu
三维不可压各向异性Navier-Stokes（ANS）方程解的大时间行为的研究是近年来才开始的。为具有完全拉普拉斯耗散的Navier-Stokes方程设计的强大工具，如傅立叶分裂方法，不再适用于只有水平耗散的情况。对于整个空间R3，当t→∞时，ANS方程的解收敛到平凡解，且收敛速度是代数的.本文研究了空间域Ω为T2×R的情形。我们的结果揭示了T2×R的大时间行为与R3的大时间行为有很大的不同。我们证明了任何小的初始速度场u 0 ∈H2（Ω）都导致唯一的整体解u在H2（Ω）中保持小。更重要的是，当t→∞时，速度场u收敛到一个非平凡的稳态。稳态的前两个分量由u 0的前两个分量的水平平均值给出，而第三个分量为零。此外，这种收敛速度是指数级的。
The study on the large-time behaviour of solutions to the 3D incompressible anisotropic Navier–Stokes (ANS) equations is very recent. Powerful tools designed for the Navier–Stokes equations with full Laplacian dissipation such as the Fourier splitting method no longer apply to the case when there is only horizontal dissipation. For the whole space R3 , as t→∞ , solutions of the ANS equations converge to the trivial solution and the convergence rate is algebraic. This paper is devoted to the case when the spatial domain Ω is T2×R . Our results reveal that the large-time behaviour for T2×R is quite different from that for R3 . We show that any small initial velocity field u0∈H2(Ω) leads to a unique global solution u that remains small in H2(Ω) . More importantly, as t→∞ , the velocity field u converges to a nontrivial steady state. The first two components of the steady state are given by the horizontal average of the first two components of u 0 while the third component vanishes. In addition, this convergence is exponentially fast.