2024/05/29 by Jacob Bedrossian, Bedrossian, Jacob, Siming He +5 · 5 citations
Engineering · Mathematics · #Stability and Controllability of Differential Equations #Computational Fluid Dynamics and Aerodynamics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2405.19249
We consider the 2D, incompressible Navier-Stokes equations near the Couette flow, ω(NS) = 1 + εω, set on the channel \mathbbT × [-1, 1], supplemented with Navier boundary conditions on the perturbation, ω|y = ± 1 = 0. We are simultaneously interested in two asymptotic regimes that are classical in hydrodynamic stability: the long time, t → ∞, stability of background shear flows, and the inviscid limit, ν→ 0 in the presence of boundaries. Given small (ε≪ 1, but independent of ν) Gevrey 2- datum, ω0(ν)(x, y), that is supported away from the boundaries y = ± 1, we prove the following results: amp; ‖ω(ν)(t) - (1)/(2π)∫ ω(ν)(t) dx ‖L2 \lesssim εe^-δν1/3 t, amp; (Enhanced Dissipation)
amp; ⟨ t ⟩ ‖u1(ν)(t) - (1)/(2π) ∫ u1(ν)(t) dx‖L2 + ⟨ t ⟩2 ‖u2(ν)(t)‖L2 \lesssim εe^-δν1/3 t, amp; (Inviscid Damping)
amp;‖ ω(ν) - ω(0) ‖L^∞ \lesssim ενt3+η, t \lesssim ν-1/(3+η) amp; (Long-time Inviscid Limit) This is the first nonlinear asymptotic stability result of its type, which combines three important physical phenomena at the nonlinear level: inviscid damping, enhanced dissipation, and long-time inviscid limit in the presence of boundaries. The techniques we develop represent a major departure from prior works on nonlinear inviscid damping as physical space techniques necessarily play a central role. In this paper, we focus on the primary nonlinear result, while tools for handling the linearized parabolic and elliptic equations are developed in our separate, companion work.