We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
On the Sobolev Stability Threshold of 3D Couette Flow in a Uniform Magnetic Field.
- Authors
Liss, Kyle
- 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 , 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 . More precisely, we show that if Re - 1 = R m - 1 ∈ (0 , 1 ] , α > 0 and N > 0 are sufficiently large, σ ∈ R \ Q satisfies a generic Diophantine condition, and the initial perturbations u in and b in to the Couette flow and magnetic field, respectively, satisfy ‖ 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 + y t , y , z) and b (t , x + y t , y , z) remain O (Re - 1) in H N ′ 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.
- Subjects
COUETTE flow; MAGNETIC fields; GLOBAL analysis (Mathematics); NAVIER-Stokes equations; SOBOLEV spaces; MAGNETOHYDRODYNAMICS; INVISCID flow; MAGNETOHYDRODYNAMIC instabilities
- Publication
Communications in Mathematical Physics, 2020, Vol 377, Issue 2, p859
- ISSN
0010-3616
- Publication type
Article
- DOI
10.1007/s00220-020-03768-3