PDE/Applied Math/Analysis Seminar

                  Friday 26. October, 1:10-2:00pm

                          6183 Bahen Center


SPEAKER: Eitan Tadmor (University of Maryland)

TITLE: Critical thresholds in Eulerian dynamics


ABSTRACT: We are concerned with the questions of global regularity vs.
finite time breakdown in Eulerian dynamics, driven  by  different models
of the nonlinear forcing. 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, O(1) critical
threshold. Our approach is based on spectral dynamics, tracing the
eigenvalues of velocity gradient which determine the boundaries of CT
surfaces in configuration space.

We demonstrate the CT phenomena with several prototype models. We begin
with the n-dimensional restricted Euler equations, obtaining a surprising
4-dimensional global existence for a large set of sub-critical initial
data. A second example consists of the corresponding n-dimensional
restricted Euler-Poisson  equations. Here we identify a set of [n/2]
spectral invariants which lead to a remarkable characterization of
two-dimensional sub-critical initial configurations with global smooth
solutions. Finally, we show how the CT phenomenon associated with
rotation prevents finite-time breakdown, which, in turn, yields a long-time
regularity regime in the shallow-water equations. Our study reveals the
critical dependence of the two-dimensional CT phenomenon on the initial
spectral gap. 

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University of Toronto PDE/Applied Math/Analysis Seminar
http://www.math.toronto.edu/appmath/

2007-2008 organizers:
          Walid Abou-Salem      walid@ math.toronto.edu
          Almut Burchard        almut@ math.toronto.edu
          Robert McCann         mccann@ math.toronto.edu