Spectral Dynamics and Critical Thresholds in Nonlinear Convective Equations

Series: 
PDE Seminar
Friday, March 13, 2009 - 16:05
1.5 hours (actually 80 minutes)
Location: 
Skiles 255
,  
University of Maryland, College Park
We discuss the global regularity vs. finite time breakdown in Eulerian dynamics, driven by different models of nonlinear forcing. Finite time breakdown depends on whether the initial configuration crosses intrinsic, O(1) critical thresholds (CT). Our approach is based on spectral dynamics, tracing the eigenvalues of the velocity gradient which determine the boundaries of CT surfaces in configuration space. We demonstrate this critical threshold phenomena with several n-dimensional prototype models. For n=2 we show that when rotational forcing dominates the pressure, it prolongs the life-span of sub-critical 2D shallow-water solutions. We present a stability theory for vanishing viscosity solutions of the 2D nonlinear "pressureless" convection. We revisit the 3D restricted Euler and Euler-Poisson equations, and obtain a surprising global existence result for a large set of sub-critical initial data in the 4D case.