Stability Threshold for 2D Shear Flows of Generalized Magnetohydrodynamic Equations Near Couette
Year: 2025
Type: article
Abstract: In this paper, we investigate the nonlinear stability threshold for the shear flows of the generalized magnetohydrodynamic (GMHD) equations on $${\mathbb {T}} \times {\mathbb {R}}$$ . We prove the nonlinear stability of the shear flow $$\left( U_s, B_s\right) =\left( (e^{-t\nu \partial _{y}^4}U(y),0)^{\top },(\alpha ,0)^{\top }\right)$$ with the initial data of the shear flow (U(y), 0) close to the Couette flow (y, 0). For sufficiently large $$|\alpha |$$ , we prove that when the initial perturbations satisfy $$\Vert (u_{in},b_{in})\Vert _{H^{N+1}}=\epsilon \ll \nu ^{\frac{7}{10}+{\tilde{\delta }}}$$ for any fixed $${\bar{\delta }}>0$$ , here $$\nu$$ is a positive real parameter, then for all $$t> 0$$ , the global solution of the 2D GMHD equations remains $$\nu ^{-\frac{1}{5}-\frac{{\bar{\delta }}}{2}}$$ $$\epsilon$$ -close to $$U_s=(e^{-t\nu \partial _{y}^4}U(y),0)^{\top }.$$
Author: Keyu Tao
Institution: Nanjing University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $1,490
OpenAlex ID: W4416613357