Regularity criteria for the Navier–Stokes equations based on one component of velocity

Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

“…When ∇u, ∇u 3 , u 3 , and the like satisfy a certain integrable condition, the weak solution is regular, and a large number of results are obtained (for details, refer to [6][7][8][9][10][11][12][13][14][15][16]). And Penel and Pokorný [13], Kukavica and Ziane [14], Cao [15], and Zhang [16], respectively, proved that the weak solution is regular on (0, T] when the weak solution satisfies the following conditions:…”

Generalizations of Troisi’s inequality in weighted p-Sobolev spaces with singularities

“…is anisotropic critical exponent and max 1≤i≤n {p i } < s. Applications of (1.1) also can be found in [3] to study the existence of fundamental solutions to anisotropic elliptic equations. Another generalization of (1.1) in [4] is used to prove regularity of the weak solution to the Navier-Stokes equations based on one component of velocity. By arithmetic and geometric mean inequality, (1.1) becomes an anisotropic Sobolev inequality presented as…”

Regularity criteria for the 3D magnetohydrodynamics system involving only two velocity components

“…However, because the flow is incompressible, one may expect regularity conditions in terms of partial components of velocity field, then there have been many criteria concerning partial components for the MHD equations [17][18][19][20] and for the Navier-Stokes equations. [21][22][23][24][25][26][27][28][29][30] Among these results, one found that the coupling effect of magnetic field cannot be neglected for these regularity conditions. Cao and Wu 4 obtained the following regularity criterion in terms of one directional derivative of…”