  We give a simplified proof of regularizing effects for first-order
Hamilton-Jacobi Equations of the form $u\_t+H(x,t,Du)=0$ in
$\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u\_t$.
As a consequence, we have a Lipschitz regularity in space and time for coercive
Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\''older
regularizing effect in space following a result of L. C. Evans and M. R. James.

Analysis of PDEs