Publications [#380034] of Kyle L Liss

Papers Published

1. Liss, K, On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field, Communications in Mathematical Physics, vol. 377 no. 2 (July, 2020), pp. 859-908 [doi]
   (last updated on 2025/06/29)

   Abstract:
   We study the stability of the Couette flow (y, 0 , 0) ^T in the presence of a uniform magnetic field α(σ, 0 , 1) on T× R× T using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit Re^{- 1}, Rm-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. More precisely, we show that if Re-1=Rm-1∈(0,1], α> 0 and N> 0 are sufficiently large, σ∈ R\ Q satisfies a generic Diophantine condition, and the initial perturbations uin and bin to the Couette flow and magnetic field, respectively, satisfy ‖uin‖HN+‖bin‖HN=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) and b(t, x+ yt, y, z) remain O(Re^{- 1}) in HN′ for some 1 ≪ N^′(σ) < N. Our proof establishes enhanced dissipation estimates describing the decay of the x-dependent modes on the timescale 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 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.