Leray-type regularizations of the Burgers and the isentropic Euler equations
- Date: 02/13/2007
Razvan Fetecau (Simon Fraser University)
University of British Columbia
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.
DG-MP-PDE Seminar 2007