Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor

Abstract

In this paper, we consider the regularity criteria for the 3D incompressible Navier-Stokes equations involving the middle eigenvalue (λ 2) of the strain tensor. It is proved that if λ + 2 belongs to Multiplier space or Besov space, then the weak solution remains smooth on [0, T ], where λ + 2 = max{λ 2 , 0}. These regularity conditions allows us to improve the result obtained by Miller [7].

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“…Recently, Miller [7] obtained another proof of (5). Later, Wu [8,9] extended (5) to the anisotropic Lebesgue spaces in the 3D double-diffusive convection equations.…”

Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor

“…implies the smoothness of the solution. More recently, the author in [27] extended the above regularity criteria to the Multiplier space and Besov space.…”

“…Recently, Miller [16] used a different method from Neustupa and Penel's to show that the following condition on the middle eigenvalue of strain tensor implies the smoothness of the solution. More recently, the author in [27] extended the above regularity criteria to the Multiplier space and Besov space.…”

The local characterizations of the singularity formation for the MHD equations

“…For incompressible MHD equations, we would like to mention here the paper of He-Xin [14] that first extended the Serrin type criteria of NS equations to the MHD equations only in terms of velocity, these results merit attention, especially, which indicates that the velocity field plays a more dominant role than the magnetic field on the regularity of solutions. For conditional regularity or blow-up criteria of NS and MHD equations, please refer to [5,6,10,19,23,27,41,42] for a more comprehensive discussions. On the other hand, He and Xin [15] introduced the definition of suitable weak solutions and obtained partial regularity theorems of suitable weak solutions.…”