Abstract:

In this talk we show that a large class of second order PDEs with smooth and bounded coefficients and Schwartz function valued initial value has a Schwartz function valued solution for all times. We gain explicit bounds on the Schwartz semi-norms by applying the Schwartz function valued version of Arzela-Ascoli to an approximate solution. These bounds show that for the Euler and the Navier-Stokes equations the vorticity (=curl of the velocity) remains a Schwartz function as long as the classical solution exists. We derive finite breakdown criteria from that. Our approach is not affected by viscosity, i.e., it treats the hyperbolic Euler and the parabolic Navier-Stokes equation in the same way. Since the solution of our class of PDEs are Schwartz functions, we can multiply with mono-/polymials and integrate to get time-depenent moments of the solution. We show that for Burgers' equation these time-dependent moments can be explicitly calculated and give a simple example how they prove a finite breakdown.