A Set-theoretic Approach to Regularity of 3D Decaying Turbulence

Abstract

This article first proves a differential version of the energy equality for the 3D decaying turbulence. It is weaker than the integral equality, as stronger conditions of continuity are needed when integrating the differential equality. Using the differential energy equality, a simple criterion for a weak solution to satisfy the strong energy inequality is proposed. Namely, the monotone-decreasing kinetic energy in time is equivalent to the LH condition. Lastly, motivated by the preceding criterion, we propose an original perspective in understanding the existing concepts of turbulence, which uses set-theoretic tools such as partial ordering and Zorn's Lemma. The partial ordering in time defines a set-theoretic characterization of regularity for weak solutions.