Archive for Rational Mechanics and Analysis

, Volume 228, Issue 3, pp 969–993 | Cite as

Global Well-Posedness of the Incompressible Magnetohydrodynamics

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Abstract

This paper studies the Cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity ν. Under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non-zero constant are sufficiently small in certain weighted Sobolev spaces, the Cauchy problem is shown to be globally well-posed for all ν ≧ 0 and all spaces with dimension n ≧ 2. Such a result holds true uniformly in nonnegative viscosity parameters. The proof is based on the inherent strong null structure of the systems introduced by Lei (Commun Pure Appl Math 69(11):2072–2106, 2016) and the ghost weight technique introduced by Alinhac (Invent Math 145(3):597–618, 2001).

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.LMNS and Shanghai Key Lab for CAM, Fudan UniversityShanghaiPeople’s Republic of China

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