GLOBAL SOLUTIONS OF THE THREE-DIMENSIONAL INCOMPRESSIBLE IDEAL MHD EQUATIONS WITH VELOCITY DAMPING IN HORIZONTALLY PERIODIC DOMAINS

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.