Blowup Criterion via Only the Middle Eigenvalue of the Strain Tensor in Anisotropic Lebesgue Spaces to the 3D Double-Diffusive Convection Equations

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

“…where q = 2p 2p−3 . For the case 3 2 < p ≤ 3, θ ∈ p, 3p 3−p , with Lemma 1.8 (15) we can estimate I as…”

“…implies the smoothness of the solution. Which was generalized in [15] to the 3D double-diffusive convection system in the anisotropic Lebesgue space. Motivated by papers cited above, we shall investigate regularity criteria of weak solutions to the 3D Navier-Stokes equations (1) in term of the middle eigenvalue of strain tensor on framework of the Multiplier space and Besov space, which extension is a generalization of methods introduced 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.…”

On the Deformation Tensor Regularity for the Navier–Stokes Equations in Lorentz Spaces

“…However, full regularity of Leary-Hopf weak solutions to the 3D Navier-Stokes system is still a fundamental open question. Starting from Serrin's famous work, regularity criteria of Leray-Hopf weak solutions are extensively studied (see [1][2][3]5,7,[9][10][11]13,[16][17][18][19][20][21][23][24][25][26]28,29] and references therein). The so-called Serrin type regularity criteria is that a weak solution u is regular on…”

“…Remark 1.1 Theorem 1.1 is an improvement of regularity criteria based on only the middle eigenvalue of the deformation tensor (1.4). It seems that a slightly modified the technique in Theorem 1.1 can be applied to other incompressible fluid equations such as Navier-Stokes equations with fractional dissipation and the double-diffusive convection equations in [28].…”