On the solution of the 4th Clay millennium problem. Equivalence of the regularity of the flow and the regularity of the deformations. Equivalence of smooth compact support and smooth Schwartz, initial conditions.

Συμβολή στην λύση του 4ου προβλήματος χιλιετίας τoυ Clay

Main Author: Kyritsis Konstantinos
Format: Article
Language: English
Published: Ο συγγραφέας 2017
Online Access: http://cris.teiep.gr/jspui/handle/123456789/1414
Physical Description: pp. 28
Abstract: In this paper I prove that the global in time regularity of the flows in the 4th Clay millennium problem, under its official formulation hypotheses, is equivalent to the corresponding regularity (bounded accumulation in finite intervals) of the deformations of the flow. It is well known that it is also equivalent to the corresponding regularity of the vorticity of the flow. We also prove, using some relatively recent ideas suggested by T. Tao , that the Schwartz initial conditions of the its official formulation of the problem in the direction of regularity are equivalent to the simpler compact support initial conditions. Finally I prove using the Helmholtz-Hodge orthogonal decomposition of vector fields, a powerful fundamental decomposition of the Euler and Navier-Stokes equations which is significant for the internal symmetries of the equations.