Stability of the 3D incompressible Navier-Stokes equations with fractional horizontal dissipation
Stability of the 3D incompressible Navier-Stokes equations with fractional horizontal dissipation
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DOI：
10.1016/j.amc.2023.127934
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
Ruihong Ji;Wen Luo;Liya Jiang
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文献类型：
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作者：
Ruihong Ji;Wen Luo;Liya Jiang
具有分数阶耗散的Navier-Stokes方程（N-S方程）的稳定性问题是数学研究的新领域之一。广义N-S方程是将N-S方程中的− Δ替换为（− Δ）α所得到的方程。已有的结果表明，当α≥ 12 + d4时，d维广义N-S方程的任何经典解在时间上总是整体的.研究了具有分数阶水平耗散（-Δ h）α的三维N-S方程的稳定性问题，其中Δ h：= Δ x12 + Δ x22.证明了对任意α∈（12，1]，H3（R3）中任意小初值所对应的解在时间上总是全局的，并且在H3（R3）中稳定.当α= 1时，有许多重要的结果。我们的结果放宽了这一要求，允许α小于1。
The stability problem of Navier–Stokes equations (N–S equations) with fractional dissipation is one of the new areas in Mathematical research. The generalized N–S equations are the equations resulting from replacing− Δ in the N–S equations by (− Δ) α. It has previously been shown that any classical solution of the d-dimensional generalized N–S equations with α≥ 1 2+ d 4 is always global in time. This paper considers the stability problem on the 3D N–S equations with only fractional horizontal dissipation (− Δ h) α, where Δ h:=∂ x 1 2+∂ x 2 2. We show that, for any α∈(1 2, 1], the solution corresponding to any sufficiently small initial data in H 3 (R 3) is always global in time and stable in H 3 (R 3). There are many important results on the case when α= 1. Our result relaxes this requirement and allows α to go below 1.