  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.
