Global small solutions to 2-D incompressible MHD system

By F. Lin, Li Xu, Ping Zhang. 2013

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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} $L^1(\R^+; Lip(\R^2))$ 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ded6c1da95d9355dc39d09a0da4bb6f85a10faf2

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Singularity formation for the incompressible Hall-MHD equations without resistivity (2013, background): “ of the background for Hall-MHD. There are many mathematical results on MHD system, for the existence of global weak solutions [9, 18], regularity criterion [11, 12] and global smooth small solutions [17, 21]. The Hall-MHD has received little attention from mathematicians. The paper [1] provided a d”
Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium (2019, background): “he nonlinear stability forthe ideal MHD equations was established inseveral beautiful papers [2,6, 19,28,34]. The stability problem for the MHD equations with no magnetic diﬀusion was ﬁrst studied in [27], which inspired many further investigations. The stability has now been successfully establishe”
Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces (2016, background): “[10] have established the existence of global-in-time solutions for initial data sufficiently close to certain equilibrium solutions (again in 2D).”
On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium (2015, background): “3) in [18, 26], @(2) y3Y there is now replaced by @(2) b0 Y , which causes new difficulty of the variable coefficients for the linearized system.”
On axially symmetric incompressible magnetohydrodynamics in three dimensions (2012, background): “ss in 2D case. A nontrivial blowup criterion for the perfect resistive MHD (1.7) was established in terms of only L1 t(BMO) norm of vorticity of the velocity ﬁeld in [10]. Recently, Lin, Xu and Zhang [11] made an important progress on the global well-posedness of classical solutions for the 2D ideal”
On Global Dynamics of Three Dimensional Magnetohydrodynamics: Nonlinear Stability of Alfvén Waves (2016, background): “For the case with strong fluid viscosity but without Ohmic dissipation, [9](see also [5]) studies the small-data-globalexistence with very special choice of data.”