Global Solutions of the Three-Dimensional Incompressible Ideal MHD Equations with Velocity Damping in Horizontally Periodic Domains
Fei Jiang, Song Jiang, Youyi Zhao
Abstract
The global existence of small smooth solutions to the equations of two-dimensional incompressible, inviscid, nonresistive magnetohydrodynamic (MHD) fluids with velocity damping has been established in [J. H. Wu, Y. F. Wu, and X. J. Xu, SIAM J. Math. Anal., 47 (2015), pp. 2630--2656]. In this paper we further study the global existence for an initial-boundary value problem in a horizontally periodic domain with finite height in three dimensions. Motivated by the multilayer energy method introduced in [Y. Guo and I. Tice, Arch. Ration. Mech. Anal., 207 (2013), pp. 459--531], we develop a new type of two-layer energy structure to overcome the difficulties arising from three-dimensional nonlinear terms in the MHD equations, and prove thus the initial-boundary value problem admits a unique global smooth solution with small initial data. Moreover, the solution decays exponentially in time to some rest state. Our two-layer energy structure enjoys two features: (1) the lower-order energy (functional) cannot be controlled by the higher-order energy; (2) under the a priori smallness assumption of the lower-order energy, we can first close the higher-order energy estimates, and then further close the lower-energy estimates in turn.