Global small solutions of 2-D incompressible MHD system

Fang-Hua Lin, Li Xu, Ping Zhang

Research output: Contribution to journalArticle

Abstract

In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical 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 a 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<sup>1</sup>(R<sup>+</sup>;Lip(R<sup>2</sup>)) estimate for the Lagrangian velocity field Y<inf>t</inf>. 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.

Original languageEnglish (US)
Pages (from-to)5440-5485
Number of pages46
JournalJournal of Differential Equations
Volume259
Issue number10
DOIs
StatePublished - Nov 15 2015

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Small Solutions
Magnetohydrodynamics
Navier Stokes equations
Global Well-posedness
Transport Equation
Lagrangian Coordinates
Change of Variables
Energy Method
Algebraic Structure
Estimate
Coupled System
Velocity Field
Navier-Stokes Equations
Regularity

Keywords

  • Anisotropic Littlewood-Paley theory
  • Dissipative estimates
  • Inviscid MHD system
  • Lagrangian coordinates

ASJC Scopus subject areas

  • Analysis

Cite this

Global small solutions of 2-D incompressible MHD system. / Lin, Fang-Hua; Xu, Li; Zhang, Ping.

In: Journal of Differential Equations, Vol. 259, No. 10, 15.11.2015, p. 5440-5485.

Research output: Contribution to journalArticle

Lin, Fang-Hua ; Xu, Li ; Zhang, Ping. / Global small solutions of 2-D incompressible MHD system. In: Journal of Differential Equations. 2015 ; Vol. 259, No. 10. pp. 5440-5485.
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