M 511

The Euler and the Navier-Stokes equations describe the motion of fluids and a smooth solution can be ensured only for small times or large viscosites. It is open if a smooth solution existence for all times or if a break down can appear in finite time. In the formulation of the problem the initial data shall be Schwartz functions. We show in arXiv:1912.11705 that then the vorticity remains a Schwartz function as long as the smooth solution exists. We provide explicit bounds for all seminorms. This provides new break down criteria for the Euler and the Navier-Stokes. Our approach is a Montel space based splitting method since the set of Schwartz functions is a Montel space. Contrary to other (especially Sobolev and weak solution) methods our approach is not affected by the viscosity, i.e., the result holds for the hyperbolic Euler and the parabolic Navier-Stokes equations.

Philipp di Dio

Leipzig