Spectral Invariants and Critical Thresholds in Eulerian Dynamics
Eitan Tadmor
Center for Scientific Computation and Mathematical Modeling (CSCAMM)
Department of Mathematics, Institute for Physical Science & Technology
University of Maryland, College Park
Abstract . We study the questions of global regularity vs. finite time breakdown
in Eulerian dynamics, ut+u·Ñxu=ÑxF,
subject to different source terms F=F(u, Ñxu,
...).
To adders 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 of the
eigenvalues, λ:=λ(Ñxu).
We shall outline three prototype cases. We begin with the n-dimensional
Restricted Euler equations, obtaining [n/2]+1 spectral invariants
which yield
surprising characterizations of critical thresholds in 3D and 4D restricted
Euler dynamics. Next we
introduce the corresponding n-dimensional Restricted Euler-Poisson (REP) system,
identifying a remarkable two-dimensional CT configurations with
global REP smooth solutions.
Finally we show how rotation prevents
finite-time breakdown.
Our study reveals the dependence of the CT phenomenon on the initial spectral gap,
λ2(0)-λ1(0).