Weak singularities of 3-D Euler equations and restricted regularity of Navier Stokes equation solutions with time dependent force terms

TL;DR

This paper constructs solutions to the 3-D Euler equations that develop weak singularities or blow up after finite time, and interprets these as solutions to Navier-Stokes equations with specially chosen time-dependent forces, revealing new insights into singularity formation.

Contribution

It introduces a novel method to construct finite-time singular solutions of the 3-D Euler equations and links these to Navier-Stokes solutions with time-dependent forces, expanding understanding of singularity mechanisms.

Findings

Existence of solutions with velocity blow-up after finite time.

Construction of weak singularities in vorticity for regular initial data.

Interpretation of Euler solutions as Navier-Stokes solutions with tailored forces.

Abstract

Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up after finite time for data in H2, and there are classical solution branches with regular data which develop weak singularities at some point of space time after finite time. The construction of these time-local solution branches is by viscosity limits of viscosity extensions of time-reversed Euler-type equations. The short time (weak) singularities are then initial values of local solution branches of the time-reversed Euler type equations, which are constructed via (not time-reversible) viscosity extensions. These solutions of the 3-D Euler equation have a straightforward interpretation as solution of incompressible Navier Stokes equation with time dependent force terms with restricted regularity or blow up of a velocity component at some point, as the time-dependent force term can be chosen such that it cancels the viscosity term of the incompressible Navier Stokes equation.