A new proof of partial regularity of solutions to Navier-Stokes equations

Abstract

In this paper we give a new proof of the partial regularity of solutions to the incompressible Navier Stokes equation in dimension 3 first proved by Caffarelli, Kohn and Nirenberg. The proof relies on a method introduced by De Giorgi for elliptic equations.

On possible isolated blow-up phenomena and regularity criterion of the 3D Navier-Stokes equation along the streamlines

“…, with A > 0 to be a given constant and δ to be a given constant with 0 < δ < 3 ) was established in [4] through applying the De Giorgi's method as developed by A. Vasseur in [17]. The main idea of the De Giorgi's method in [17] is based on the establishment of the following nonlinear recurrence relation of the energy…”

“…The main idea of the De Giorgi's method in [17] is based on the establishment of the following nonlinear recurrence relation of the energy…”

“…+ of the solution u to (1.1) over a certain space time region (for a precise definition of U k , see section 3 of this paper, or alternatively [17] or [4]),…”

“…According to the idea in [17], for a given solution u to (1.1) on [0, T ) × R 3 with possible blow up time T , the L ∞ -boundedness conclusion |u| R over […”

“…Roughly speaking, λ > 0 ensures the smallness of the energy U 1 of the first truncated function v 1 , due to the fact that 1 R λ will become small as R is sufficiently large. The smallness of U 1 will trigger the nonlinear recurrence effect of relation (1.2) which eventually causes the very fast decay of U k to 0 (see Lemma 3.2 which originally appeared in [17] ). This resulting decay of U k to 0 then implies the desired boundedness conclusion |u| R over […”