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

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