Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

Abstract

In this paper, we solve a long-standing open problem: nonlinear stability of the current-vortex sheet in the ideal incompressible magnetohydrodynamics under the Syrovatskij stability condition. This result gives the first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability. © 2017 Wiley Periodicals, Inc. the solution u˙WD uj ṫ ; h˙WD hj ṫ ; p˙WD pj ṫ ;

Well-posedness of the free boundary problem in compressible elastodynamics

“…the operators L ′ e and B ′ e are defined in (33) and (34), and (U n+1/2 , ϕ n+1/2 ) is a smooth modified state such that (U a + U n+1/2 , ϕ a + ϕ n+1/2 ) satisfies constraints (25)- (27) and (41) (Ψ n , Ψ n+1/2 , and δΨ n are associated to ϕ n , ϕ n+1/2 , and δϕ n like Ψ is associated to ϕ). The right-hand sides f n and g n are defined through the accumulated errors at the step n.…”

“…Wang and Zhang for current-vortex sheets [33] and the plasma-vacuum problem [34] in ideal incompressible magnetohydrodynamics (see also [43] mentioned above). At the same time, in the present paper our result connected with Rayleigh-Taylor instability detected as ill-posedness for frozen coefficients is also obtained for the incompressible case.…”

Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition

“…The nonlinear stability of compressible current-vortex sheets was solved independently by Chen and Wang [3] and Trakinin [29] by using the Nash-Moser iteration. For incompressible current-vortex sheets, Coulombel, Morando, Secchi and Trebeschi [6] proved an a priori estimate for the nonlinear problem under a strong stability condition, and Sun, Wang and Zhang [26] solved the nonlinear stability very recently.…”

“…While for the case of incompressible elastic fluid with constant density it was studied by Hao and Wang in [14] where a priori estimates in Sobolev norms of solutions were derived through a geometrical point of view of [5] under the fulfilment of the Rayleigh-Taylor sign condition (1.13). Very recently, Li, Wang and Zhang [18] estabilished the local well-posedness for both two free boundary problems in incompressible elastodynamics under a natural stability condition by using the method developed in [26]. For elastodynamics equations, the elasticity plays a stabilization role, see, for instance [4,31].…”

Ill-Posedness of Free Boundary Problem of the Incompressible Ideal MHD

“…It was found by Ebin [6] that the free boundary problem for incompressible Euler equations is ill-posed when the Taylor sign condition fails. A natural question for MHD free boundary problem is that if this is still the case or the magnetic field has some stablizing effect, as discussed for the current-vortex sheet problem of MHD in [3,23,25]. A stability condition,…”