We study unique continuation properties of solutions of
linear and non-linear Schroedinger equations. In the nonlinear case we are interested in deducing uniqueness of the solution from information on the difference of two possible solutions at two different times.

This talk will give an introduction to optimal regularity as a tool to analyze (fully) nonlinear parabolic equations/systems. After a review of the major developments of the theory, the focus will shift to singular parabolic equations. It will be shown that optimal regularity results can be obtained for a large class of singular abstract Cauchy problems and, if time permits, applications of the theory will be presented.

We will analyze the physiological and psychophysical mechanisms underlying nonlinear processing in the auditory system. We will present a nonlinear model in the amplitude-frequency domain (Companding) and itsimplementation in an attempt to improve speech recognition in noise. We will examine key parameters in the model and their behavioral relevance in terms of functional gain in both normal-hearing and cochlear-implant listeners.

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length.
To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows.

This talk will discuss results on singularity formation for two different fluid flow problems,
and the physical significance of these results.

The first problem is the so-called Muskat problem, which describes the evolution
of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell. In contrast to the Hele-Shaw problem (the one phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem.

For the stable Muskat problem, in which the higher viscosity fluid expands into the lower-viscosity
fluid, it is shown that smooth solutions exist for all t>0, even if the initial data contains weak (e.g., curvature) singularities.

For the unstable problem, in which the higher viscosity fluid contracts, solutions are constructed that start
off smooth but develop a singularity in finite time.

The second example is the unsteady Prandtl equations for flow in an incompressible boundary layer.
A semi-analytic method for constructing singular solutions is suggested, and some preliminary results
toward this construction are presented.

This is joint work with Russ Caflisch and Sam Howison.