Regularity criteria of the 4D navier-stokes equations involving two velocity field components

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Abstract

We study the Serrin-type regularity criteria for the solutions to the four-dimensional Navier-Stokes equations and magnetohydrodynamics system. We show that the sufficient condition for the solution to the four-dimensional Navier-Stokes equations to preserve its initial regularity for all time may be reduced in the following ways: from a bound on the four-dimensional velocity vector field to any two of its four components; from a bound on the gradient of the velocity vector field to the gradient of any two of its four components; and from a gradient of the pressure scalar field to any two of its partial derivatives. Results are further generalized to the magnetohydrodynamics system. These results may be seen as a four-dimensional extension of many analogous results that exist in the three-dimensional case and also component reduction results of many classical results.

Original languageEnglish
Pages (from-to)2229-2252
Number of pages24
JournalCommunications in Mathematical Sciences
Volume14
Issue number8
DOIs
StatePublished - 2016

Keywords

  • Magnetohydrodynamics system
  • Navier-Stokes equations
  • Regularity criteria
  • Scaling-invariance

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