The Global Well-Posedness and Decay Estimates for the 3D Incompressible MHD Equations With Vertical Dissipation in a Strip

Abstract

The three-dimensional incompressible magnetohydrodynamic (MHD) system with only vertical dissipation arises in the study of reconnecting plasmas. When the spatial domain is the whole space $\mathbb R^3$, the small data global well-posedness remains an extremely challenging open problem. The one-directional dissipation is simply not sufficient to control the nonlinearity in $\mathbb R^3$. This paper solves this open problem when the spatial domain is the strip $\Omega := \mathbb R^2\times [0,1]$ with Dirichlet boundary conditions. By invoking suitable Poincaré type inequalities and designing a multi-step scheme to separate the estimates of the horizontal and the vertical derivatives, we are able to establish the global well-posedness in the Sobolev setting $H^3$ as long as the initial horizontal derivatives are small. We impose no smallness condition on the vertical derivatives of the initial data. Furthermore, the $H^3$-norm of the solution is shown to decay exponentially in time. This exponential decay is surprising for a system with no horizontal dissipation. This large-time behavior reflects the smoothing and stabilizing phenomenon due to the interaction within the MHD system and with the boundary.

Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation

“…Since then, numerous works have been devoted to this issue, and stability for the 2D MHD system with various partial dissipations has been successfully established (see, e.g. [1,3,4,8,11,15,16,18,19,22,32,33,35,36,39,41,43,45,[50][51][52]). For the 2D magneto-micropolar fluid equations (1.2), global regularity has been obtained by Cheng and Liu [5].…”