Global small solutions to 2-D incompressible MHD system

Abstract

In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with small and smooth initial data. 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 due to the fact that there is a free transport equation in the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system \eqref{1.1} in the Lagrangian coordinates \eqref{a14}. Then we employ anisotropic Littlewood-Paley analysis to establish the key {\it a priori} $L^1({\mathbb{R}}^{+};Lip({\mathbb{R}}^2))$ estimate to the Lagrangian velocity field $Y_t$. With this estimate, we prove the global wellposedness of \eqref{a14} with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of \eqref{a14} is crucial for the proofs to work. The global wellposedness of the original system \eqref{1.1} then follows by a suitable change of variables.

Cited by in corpus (15)

• Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes
• Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces
• Large time behavior of solutions to 3-D MHD system with initial data near equilibrium
• Finite time singularity of the nematic liquid crystal flow in dimension three
• An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system
• Global Existence for Two Dimensional Incompressible Magnetohydrodynamic Flows with Zero Magnetic Diffusivity
• Global small solutions to three-dimensional incompressible MHD system
• Global smooth solutions of MHD equations with large data
• The 2D Incompressible Magnetohydrodynamics Equations with only Magnetic Diffusion
• Well-posedness for compressible MHD system with highly oscillating initial data
• Singularity formation for the incompressible Hall-MHD equations without resistivity
• On the Yudovich solutions for the ideal MHD equations
• Global small solution to the 2D MHD system with a velocity damping term
• Global solvability, non-resistive limit and magnetic boundary layer of the compressible heat-conductive MHD equations
• Well-Posedness and Optimal Time-Decay for Compressible MHD System in Besov Space