当前位置: X-MOL 学术Commun. Math. Phys. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field
Communications in Mathematical Physics ( IF 2.4 ) Pub Date : 2020-06-11 , DOI: 10.1007/s00220-020-03768-3
Kyle Liss

We study the stability of the Couette flow $$(y,0,0)^T$$ ( y , 0 , 0 ) T in the presence of a uniform magnetic field $$\alpha (\sigma , 0, 1)$$ α ( σ , 0 , 1 ) on $${{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}$$ T × R × T using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit $$\mathbf{Re} ^{-1}$$ Re - 1 , $$\mathbf{R }_m^{-1} \ll 1$$ R m - 1 ≪ 1 and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations small in the Sobolev space $$H^N$$ H N . More precisely, we show that if $$\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]$$ Re - 1 = R m - 1 ∈ ( 0 , 1 ] , $$\alpha > 0$$ α > 0 and $$N > 0$$ N > 0 are sufficiently large, $$\sigma \in {{\mathbb {R}}}{\setminus } {\mathbb {Q}}$$ σ ∈ R \ Q satisfies a generic Diophantine condition, and the initial perturbations $$u_{\text{ in }}$$ u in and $$b_{\text{ in }}$$ b in to the Couette flow and magnetic field, respectively, satisfy $$\Vert u_{\text{ in }}\Vert _{H^N} + \Vert b_{\text{ in }}\Vert _{H^N} = \epsilon \ll \mathbf{Re} ^{-1}$$ ‖ u in ‖ H N + ‖ b in ‖ H N = ϵ ≪ Re - 1 , then the resulting solution to the 3D MHD equations is global in time and the perturbations $$u(t,x+yt,y,z)$$ u ( t , x + y t , y , z ) and $$b(t,x+yt,y,z)$$ b ( t , x + y t , y , z ) remain $${\mathcal {O}}(\mathbf{Re} ^{-1})$$ O ( Re - 1 ) in $$H^{N'}$$ H N ′ for some $$1 \ll N'(\sigma ) < N$$ 1 ≪ N ′ ( σ ) < N . Our proof establishes enhanced dissipation estimates describing the decay of the x -dependent modes on the timescale $$t \sim \mathbf{Re} ^{1/3}$$ t ∼ Re 1 / 3 , as well as inviscid damping of the velocity and magnetic field with a rate that agrees with the prediction of the linear stability analysis. In the Navier–Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption $$\epsilon \ll \mathbf{Re} ^{-3/2}$$ ϵ ≪ Re - 3 / 2 (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017). The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier–Stokes equations linearized around Couette flow.

中文翻译:

均匀磁场中 3D 库埃特流的 Sobolev 稳定性阈值

我们研究了在均匀磁场 $$\alpha (\sigma , 0, 1)$ 存在下 Couette 流 $$(y,0,0)^T$$ ( y , 0 , 0 ) T 的稳定性$ α ( σ , 0 , 1 ) 在 $${{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}$$ T × R × T使用 3D 不可压缩磁流体动力学 (MHD) 方程。我们考虑无粘性的理想导体极限 $$\mathbf{Re} ^{-1}$$ Re - 1 , $$\mathbf{R }_m^{-1} \ll 1$$ R m - 1 ≪ 1并证明对于强且适当定向的背景场,Couette 流对于 Sobolev 空间 $$H^N$$HN 中的小扰动是渐近稳定的。更准确地说,我们证明如果 $$\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]$$ Re - 1 = R m - 1 ∈ ( 0 , 1 ] , $$\alpha > 0$$ α > 0 和 $$N > 0$$ N > 0 足够大,我们的证明建立了增强的耗散估计,描述了时间尺度上 x 依赖模式的衰减 $$t \sim \mathbf{Re} ^{1/3}$$ t ∼ Re 1 / 3 ,以及无粘性阻尼速度和磁场的速率与线性稳定性分析的预测一致。在 Navier-Stokes 的情况下,只有在更强的假设下才知道对适应 Couette 流混合的坐标系中的扰动的高规律性控制 $$\epsilon \ll \mathbf{Re} ^{-3/2} $$ ϵ ≪ Re - 3 / 2 (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017)。MHD 设置的改进是可能的,因为磁场引起时间振荡,部分抑制了提升效应,这是围绕 Couette 流线性化的 Navier-Stokes 方程的主要瞬态增长机制。
更新日期:2020-06-11
down
wechat
bug