(Sept. 28) R. J. McCANN Sharp decay rates and next asymptotics for fast nonlinear diffusion---
ABSTRACT: In many diffusive settings, initial disturbances will gradually disappear and all but their crudest features --- such as size and location --- will eventually be forgotten. Quantifying the rate at which this information is lost is sometimes a question of central interest. Joint works with Yong Jung Kim (KAIST) and Dejan Slepcev (UCLA) address this issue for the conservative nonlinearities in a model problem known as the fast diffusion equation $$ u_t = \Delta(u^m), \qquad (n-2)_+/n < m <1 \quad u,t \ge 0, \quad x \in {\bf R}^n, $$ which governs the decay of any integrable, compactly supported initial density towards a characteristically spreading self-similar profile. For other values of the parameter $m$, this equation has been used to model heat transport, population spreading, fluid seepage, curvature flows, and avalanches in sandpiles.
For the fastest conservative nonlinearities $m \le n/(n+2)$, we develop a potential theoretic comparison technique with Kim which establishes the sharp conjectured power law rate of decay $1/t$ uniformly in relative error, and in weaker norms such as $L^1({\bf R}^n)$. For nonlinearities $m \ge (n-1)/n$ we attain nearly the same $L^1({\bf R}^n)$ rate as a second order asymptotic after centering the mass of the solution, using an entropy dissipation approach with Slepcev. This leaves a gap in dimensions $n \ge 3$.

(Oct. 5) Walter CRAIG Global solutions for the Boltzmann equation in the shock tube geometry ---
ABSTRACT: I will discuss solutions of the Boltzmann equation which have bounded initial density, for which the total energy and entropy are controlled. In the shock tube geometry (in which solutions depend upon three velocity variables but only one space variable) we show that the density remains bounded for all finite times. Among several corollaries, the result implies the uniqueness of dissipative weak solutions in the sense of P.-L. Lions. The main theorem depends upon a newly described averaging property of the Boltzmann equation.

(Oct. 12) Joseph CONLON An Optimal Control Problem in Insurance and Banking ---
ABSTRACT: In this talk we will discuss a model for the efficient operation of a bank or insurance company which was recently proposed by Peura and Keppo. In the model the bank invests its capital in a risky asset described by Brownian motion with a positive drift. The bank also pays dividends to its owners and raises capital from them. The bank is considered to be operating most efficiently if the expected net payments to the owners over the lifetime of the bank is maximum. This gives rise to an optimal control problem which we show has a unique solution. The main mathematical issue in the proof is the fact that oscillations of solutions to the heat equation decrease with time. This is joint work with Hyekyung Min.

(Oct. 19) Virginia YOUNG Financial valuation of mortality risk: Pricing and hedging under stochastic hazard rates ---
ABSTRACT: We develop a theory for pricing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing the mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. The resulting price solves a non-linear parabolic differential equation, and the price satisfies a number of desirable properties. In this talk, we focus on deriving the limiting value of the price per risk as the number of insured risks goes to infinity.

(Oct. 26) Luca CAPOGNA Generalized mean curvature flow in Carnot groups ---
ABSTRACT: In joint, ongoing work with Giovanna Citti (U. Bologna) we study weak solutions of the mean curvature flow in a Carnot group, using an analogue of the level set method (as in the work of Evans-Spruck, Chen-Giga-Goto, etc etc). The difficulties in the new setting arise from the non-commutativity of the derivates and from the presence of characteristic points. We prove existence of weak solutions as limit of the corresponding Riemannian problems, in a penalization approximation scheme, and establish comparison principles.

(November 2) Yann BERNARD Gauge fields in curved space-times ---
ABSTRACT: We consider a Dirac particle with N internal degrees of freedom on a static spherically symmetric 4-dimensional Lorentzian manifold, and derive its equation of motion by coupling Eintein's Fields equations to the Dirac equation and an SU(N) Yang-Mills system. The approach is new, and we will derive a condition relating the SU(N)-induced Yang-Mills connection on the base space to its "natural" Levi-Civita connection. We will then focus on a gauge field U(1)xSU(2), and state non-existence as well as regularity results for the corresponding EDYM system.

(November 7) Alina CHERTOCK Effects of Saturating Diffusion on Strongly Degenerate Parabolic Equations ---
ABSTRACT: We consider a nonlinear degenerate parabolic equation used to describe propagation of thermal waves in plasma or in a porous medium, endowed with a mechanism for saturation of the diffusion flux which corrects the nonphysical gradient-flux relations at high gradients. We study the model both analytically and numerically, and discover that in certain cases the motion of the front is controlled by the saturation mechanism and instead of the typical infinite gradients resulting from the linear flux-gradients relations we obtain a sharp, shock-like front, typically associated with nonlinear hyperbolic phenomena. We prove that if the initial support is compact, independently of the smoothness of the initial datum inside the support, a sharp front discontinuity forms in a finite time, and until then the front does not expand. Adding a nonlinear convection enhances the conditions for a breakdown. In fact, the most interesting feature is the effect of criticality, that is, unlike small amplitude solutions that remain smooth at all times, large amplitude solutions may develop discontinuities. This feature is easily seen via the analysis of traveling waves (TW): while small amplitude kinks are smooth, in large amplitude kinks part of the upstream-downstream transition must be accomplished via a discontinuous jump (subshocks). Thus induced discontinuities may persist indefinitely since the TWs represent a forced motion. Unlike the classical Burgers case, here, due to the saturation of fluxes, the viscous forces have a bounded range. When the inertial forcing exceeds a certain threshold, the disparity between the inertial and dissipative forces is resolved by formation of a discontinuity. We also demonstrate that equation presents a real numerical challenge - in some cases, the jumps are quite small, and they may decrease further with time. In order to resolve "small" discontinuities, one needs to achieve a superior resolution in the neighborhoods of the shocks. This is a joint work with A. Kurganov and P. Rosenau

(November 9) Tiffany SHAW Multiscale Mixing Efficiencies for Steady Sources ---
ABSTRACT:We study the evolution of a passive scalar maintained by steady sources and sinks, and stirred by an incompressible flow field. The effectiveness of a given flow to enhance mixing over molecular diffusion on various length scales may be quantitatively measured by the suppression of the space-time averaged scalar variance, gradient variance, and inverse gradient variance. The ratio of each variance without stirring to the corresponding variance with stirring provide non-dimensional measures of the "mixing efficiency" of the flow on different scales. Here we report rigorous estimates on these multiscale mixing efficiencies for flows including statistically homogeneous and isotropic stirring fields, and compare them with direct numerical simulations and exact calculations for a sample problem.

(November 16) Gunilla KREISS Interface capturing methods for multi-phase flows ---
ABSTRACT: There are 2 main categories of interface capturing methods developed for the simulation of multiphase flow, level set methods and phasefield methods. In this talk we focus on methods useful for simulating material processes such as solidification and sintering. To begin with we discuss a new conservative level set method for incompressible two-phase flow. The method is conservative in the sense that no mass is lost. In cases the inner structure and dynamics of the interface are important, for example when segregation of solutes to grain boundaries occurs, a phasefield method based on the Cahn-Hilliard theory of diffuse interfaces is more appropriate. Examples will be given wetting phenomena in fluids and diffusion controlled phase change in materials.

(November 30) Lorendana Lanzani ---Generalized Gagliardo-Nirenberg inequalities and Hodge systems in R^n ---
ABSTRACT: We show how a natural generalization to forms of the classical Gagliardo-Nirenberg inequality (and of another inequality recently proved by Bourgain and Brezis) yields a-priori L^p estimates for L^1_loc solutions of div-curl and, more generally, the Hodge system with data in L^1(R^n). This work is joint with E. M. Stein.

(December 1) Yan GUO --- On stable galaxy models ---
ABSTRACT: The dynamics of a galaxy can be described by the Vlasov-Poisson system, where a large number of stars interact through a self-consistent gravitational field collectively. There are many important steady galaxy configurations in such a model, and their dynamical stability has been an important question in astrophysics. In this talk, we will review recent nonlinear stability results for these steady galaxy models from a mathematical standpoint.

(December 7) Jeffrey RAUCH ---Short wavelength asymptotics for hyperbolic internal waves ---
ABSTRACT: With Olivier Gues, the folded boundaary value problem method developed to study boundary layers and shock structure has been imported and refined to study the propagation of internal layers. An example of an internal layer is a pulse in which case the values on the two sides of the layer agree. Alterman-Rauch treated pulses in which case they could justify one term of an asyptotic expansion and their method failed for internal layers. The new technique yields infinitly accurate expansions in both cases. The diffraction of internal layers remains open.