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

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.