Schwartz Function Valued Solutions of the Euler and the Navier-Stokes Equations

Abstract

We prove the existence of a solution for the second order system of partial differential equations ${\partial }_{t}f=\nu \cdot \Delta f+g\cdot \nabla f+h\cdot f+k$ by a Montel space version of Arzelà--Ascoli and bound all Schwartz semi-norms. We find that for the Euler and the Navier--Stokes equations the vorticity remains a Schwartz function as long as the classical solution exists. Our approach is not affected by viscosity. It treats the hyperbolic Euler and the parabolic Navier--Stokes equation simultaneously.

arXiv commentlength shortened, misprints corrected, 19 pages + 4 pages literature