Leray-type regularizations of the Burgers and the isentropic Euler equations

We start from the Burgers equation vt + vvx = 0 and investigate a smoothing mechanism that replaces the convective velocity v in the nonlinear term by a smoother velocity field u. This type of regularization was first proposed in 1934 by Leray, who applied it in the context of the incompressible Navier-Stokes equations. We show strong analytical and numerical indication that the Leray smoothing procedure yields a valid regularization of the Burgers equation. We also study the stability of the front traveling waves. The front stability results show that the regularized equation mirrors the physics of rarefaction and shock waves in the Burgers equation. Finally, we apply the Leray regularization to the isentropic Euler equations and use the weakly nonlinear geometrical optics (WNGO) asymptotic theory to analyze the resulting system. As it turns out, the Leray procedure regularizes the Euler equations only in special cases. We further investigate these cases using Riemann invariants techniques.