## UC Berkeley / Lawrence Berkeley Laboratory

#### Spectral Dynamics and Critical Thresholds in Nonlinear Convective Equations

**Eitan Tadmor, University of Maryland**

##### March 18th, 2010 at 2PM–3PM in 740 Evans Hall [Map]

We discuss the global regularity vs. finite time breakdown in nonlinear
convection driven by different models of 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 we obtain a surprising global existence
result for a large set of sub-critical initial data in the 4D case.