Global small solutions of 2-D incompressible MHD system

Abstract

Abstract. In this paper, we consider the global wellposedness of 2-D incompressible magnetohydrodynamical system with smooth initial data which is close to some non-trivial steady state. It is a coupled system between the Navier-Stokes equations and a free transport equation with an universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system (1.1) in the Lagrangian coordinates (2.19). Then we employ anisotropic Littlewood-Paley analysis to establish the key a priori L 1 (R + ; Lip(R 2 )) estimate for the Lagrangian velocity field Yt. With this estimate, we can prove the global wellposedness of (2.19) with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of (2.19) is crucial for the proofs to work. The global wellposedness of the original system (1.1) then follows by a suitable change of variables.

Finite Time Singularity of the Nematic Liquid Crystal Flow in Dimension Three

“…The system (1.1) for (u, d) is closely related to the MHD system for (u, ψ) provided we identify ψ = ∇ × d. There have been many interesting works on global small solutions to the MHD system recently. See, for example, Lin-Zhang [32] and Lin-Zhang [33] for n = 3 and Lin-Zhang-Xu [34] for n = 2.…”

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

“…Jiu and Niu [7] established the local existence of solutions in 2D for initial data in H s for integer s 3, and more recently Ren et al [12] and Lin et al [10] have established the existence of global-in-time solutions for initial data sufficiently close to certain equilibrium solutions (again in 2D).…”

Decoupled schemes for unsteady MHD equations II: Finite element spatial discretization and numerical implementation

“…To our best knowledge, the mathematical research for MHD equations originated from the paper [4] by M. Sermange, R. Temam. Up to now, there are many articles discussing MHD equations from theoretical view points, please refer [5][6][7][8][9][10][11] and their references. At the same time, numerical study for MHD equations thrives in the recent years.…”