On global regularity and singularities of Navier-Stokes- and Euler equation solutions

TL;DR

This paper investigates the conditions under which solutions to the Navier-Stokes and Euler equations remain regular or develop singularities, introducing new data functions and conditions that influence global regularity and blow-up phenomena.

Contribution

It defines Euler-Leray data functions and establishes their Lipschitz continuity as sufficient for global regularity bounds, also exploring conditions leading to vorticity blow-ups and multiple solutions.

Findings

Lipschitz continuity of Euler-Leray data functions ensures global regular bounds.

Strong polynomial decay of initial data prevents singularities in Euler solutions.

Lipschitz conditions can lead to vorticity blow-ups and multiple solutions.

Abstract

Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the Cauchy data satisfy an additional condition of strong polynomial decay at spatial infinity. Furthermore, if a Lipschitz condition for the Euler- Leray data function of second order is satisfied, then there are long time vorticity blow ups of the incompressible Euler equation, and, correspondingly, short time and long time vorticity blow ups or singular solutions of incompressible Navier Stokes equations with time-dependent forces of lower regularity. A further consequence is that multiple global solutions of the incompressible Euler equations exists.