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

TL;DR

This paper demonstrates that solutions to the Euler and Navier-Stokes equations preserve Schwartz function properties over time, using a novel approach that applies to both hyperbolic and parabolic PDEs without being affected by viscosity.

Contribution

It introduces a method to prove the persistence of Schwartz class solutions for both Euler and Navier-Stokes equations, independent of viscosity effects.

Findings

Vorticity remains a Schwartz function as long as classical solutions exist.

The approach applies simultaneously to hyperbolic and parabolic PDEs.

Solutions are bounded in all Schwartz semi-norms.

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 Arzel\`a--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.