Regularity of weak solutions to 3D incompressible Navier–Stokes equations

Abstract

In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier-Stokes equations in the class of L r (0, T ; L 3 ( )) with r ∈ [1, ∞), which are beyond Serrin's condition.

Interior regularity of weak solutions to the perturbed Navier-Stokes equations

“…In the three dimensional case, a large gap remains between the regularity available in the existence results and the additional regularity required in the sufficient conditions to guarantee the smoothness of weak solutions of the standard Navier-Stokes equations. This gap has been narrowed by the works of Iskauriaza-Seregin-Sverak [18], LadayzhenskayaSeregin [19], Scheffer [25], Serrin [27], Struwe [29], see also [2], [3], [4], [5], [6], [8], [9], [10], [13], [14], [15], [16], [20], [22], [23], [24], [26], [31], [32] and the references therein, which bring about a deeper understanding of the regularity. In particular, some local partial regularity results and Hausdorff dimension estimates on the possible singular set have been obtained for a class of suitable weak solutions defined and constructed in [7], where the principal tools are the so-called generalized energy inequality and a scaling argument.…”

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

“…Many mathematicians have made their contribution to the regularity theory. And various kinds of regularity criteria have been obtained for weak solutions to the incompressible Navier-Stokes equations in [15][16][17][18][19][20][21][22][23][24][25][26][27][28] and the references therein. In [29][30][31][32], Scheffer began to develop the analysis about the possible singular points set and establish various partial regularity results for a class of weak solutions.…”