Global regularity of logarithmically supercritical MHD system with improved logarithmic powers

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Abstract

The magnetohydrodynamics system consists of a coupling of the Navier-Stokes equations and Maxwell’s equation from electromagnetism. We extend the work of [2] on the Navier-Stokes equations to the magnetohydrodynamics system to prove its global well-posedness with logarithmically supercritical dissipation and diffusion with the logarithmic power that is improved in contrast to the previous work of [14]. The main difficulty is that the method in [2] relies heavily on the symmetry within the Navier-Stokes equation, which is lacking in the magnetohydrodynamics system due to the non-linear terms that are mixed with both velocity and magnetic fields; this difficulty may be overcome by somehow taking advantage of the symmetry within the energy formulation of the magnetohydrodynamics system appropriately.

Original languageEnglish
Pages (from-to)147-173
Number of pages27
JournalDynamics of Partial Differential Equations
Volume15
Issue number2
DOIs
StatePublished - 2018

Keywords

  • Fractional Laplacians
  • Global regularity
  • Magnetohydrodynamics system
  • Navier-Stokes equations
  • Supercritical

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