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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2004-2005 MATHEMATICS COLLOQUIUM SERIES 01/12/05 Colloquium Dr. Razvan C. Fetecau Szegö Assistant Professor Mathematics Department Stanford University The Lagrangian Averaging for Compressible Fluids Abstract:   We first present an extension of the Lagrangian averaged Euler (LAE - a) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Two such regularizations of 1D compressible Euler equations are considered. Both regularizations are derived from a variational principle that is a compressible generalization of that used to derive the Camassa-Holm equation. We show, using dynamical systems methods, that both systems have families of solitary wave solutions in both density and velocity. The interaction of the solitary waves, under both models, is studied numerically using an energy-preserving method. Motivated by the results obtained with the compressible models considered above, we study a family of nonlinear dispersive regularizations of the Burgers equation. We investigate the regularity and the zero-dispersion limit of the solutions of some PDE's in this family. Dr. Fetecau received the Ph.D. in 2003 from the Department of Applied and Computational Mathematics at California Institute of Technology. Presently, he is a Szegö Assistant Professor in the Mathematics Department at Stanford University