CMU Campus
Center for                           Nonlinear Analysis
CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact
Seminar Abstracts

Eitan Tadmor, Center for Scientific Computation and mathematical Modeling (CSCAMM), Department of Mathematics and Institute for Physical Science & Technology, University of Maryland.

"Critical Thresholds in Eulerian Dynamics "

Abstract

 

We study the questions of global regularity vs. finite time breakdown in Eulerian dynamics, $u_t + u \cdot \nabla u = \nabla F$, which shows up in different contexts dictated by modeling of $F$'s. To address these questions, we propose the notion Critical Threshold (CT), where a conditional finite time breakdown depends on whether the initial configuration crosses an intrinsic, ${\cal O}(1)$ critical threshold. Our approach is based on a main new tool of spectral dynamics, where the eigenvalues, $\lambda := \lambda (\nabla u)$, and eigenpairs $(\ell,r)$, are traced b y the forced Raccati equation $\lambda_t + u
\cdot \nabla \lambda + \lambda^2 =\left<\ell,D^2Fr\right>$. We shall outline three prototype cases.

We begin with the $n$-dimensional Restricted Euler equations, obtaining $[n/2]+1$ global invariants which precisely characterize the local topology at breakdown time. Next we introduce the corresponding $n$-dimensional Restricted Euler-Poisson (REP) system, identifying a set of $[n/2]$ global invariants, which yield (i) sufficient conditions for finite time breakdown, and (ii) a remarkable characterization of two-dimensional initial REP configurations with global smooth solutions. And finally, we show that a CT phenomenon associated with rotation prevents finite-time breakdown. Our study reveals the dependence of the CT phenomenon on the initial spectral gap, $\lambda_2(0)-\lambda_1(0)$.

THURSDAY, April 1, 2004
Time: 1:30 P.M.
Location: PPB 300