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Series: PDE Seminar

The problem of understanding the parabolic hull of Brownian motion arises in two different fields. In mathematical physics this is the Burgers-Hopf caricature of turbulence (very interesting, even if not entirely turbulent). In statistics, the limit distribution we study was first considered by Chernoff, and forms the cornerstone of a large class of limit theorems that have now come to be called 'cube-root-asymptotics'. It was in the statistical context that the problem was first solved completely in the mid-80s by Groeneboom in a tour de force of hard analysis. We consider another approach to his solution motivated by recent work on stochastic coalescence (especially work of Duchon, Bertoin, and my joint work with Bob Pego). The virtues of this approach are simplicity, generality, and the appearance of a completely unexpected Lax pair. If time permits, I will also indicate some tantalizing links of this approach with random matrices. This work forms part of my student Ravi Srinivasan's dissertation.

Series: PDE Seminar

We study generalized traveling front solutions of reaction-diffusion equations modeling flame propagation in combustible media. Although the case of periodic media has been studied extensively, until very recently little has been known for general disordered media. In this talk we will address questions of existence, uniqueness, and stability of traveling fronts in this framework.

Series: PDE Seminar

The transportation problem can be formulated as the problem of finding the optimal way to transport a given measure into another with the same mass. In mathematics, there are at least two different but very important types of optimal transportation: Monge-Kantorovich problem and ramified transportation. In this talk, I will give a brief introduction to the theory of ramified optimal transportation. In terms of applied mathematics, optimal transport paths are used to model many "tree shaped" branching structures, which are commonly found in many living and nonliving systems. Trees, river channel networks, blood vessels, lungs, electrical power supply systems, draining and irrigation systems are just some examples. After briefly describing some basic properties (e.g. existence, regularity) as well as numerical simulation of optimal transport paths, I will use this theory to explain the dynamic formation of tree leaves. On the other hand, optimal transport paths provide excellent examples for studying geodesic problems in quasi-metric spaces, where the distance functions satisfied a relaxed triangle inequality: d(x,y) <= K(d(x,z)+d(z,y)). Then, I will introduce a new concept "dimensional distance" on the space of probability measures. With respect to this new metric, the dimension of a probability measure is just the distance of the measure to any atomic measure. In particular, measures concentrated on self-similar fractals (e.g. Cantor set, fat Cantor sets) will be of great interest to us.

Series: PDE Seminar

In this talk I will present Hamiltonian identities for elliptic PDEs and systems of PDEs. I will also show some interesting applications of these identities to problems related to solutions of some nonlinear elliptic equations in the entire space or plane. In particular, I will give a rigorous proof to the Young's law in triple junction configuration for a vector-valued Allen Cahn model arising in phase transition; a necessary condition for the existence of certain saddle solutions for Allen-Cahn equation with asymmetric double well potential will be derived, and the structure of level sets of general saddle solutions will also be discussed.

Series: PDE Seminar

Image segmentation has been widely studied, specially since Mumford-Shah functional was been proposed. Many theoretical works as well as numerous extensions have been studied rough out the years. In this talk, I will focus on couple of variational models for multi-phase segmentation. For the first model, we propose a model built upon the phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. For the second model, we propose a variational functional for an unsupervised multiphase segmentation, by adding scale information of each phase. This model is able to deal with the instability issue associated with choosing the number of phases for multiphase segmentation.

Series: PDE Seminar

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by Darwin. In a very simple, general, and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born population undergoes small variance on the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait?
We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'best fitted trait' and eventually compute various forms of branching points, which represent the cohabitation of two different populations.
The concepts are based on the asymptotic analysis of the above mentioned parabolic equations, one appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed.
This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G. Barles.

Series: PDE Seminar

We calculate numerically the solutions of the stationary Navier-Stokes equations in two dimensions, for a square domain with particular choices of boundary data. The data are chosen to test whether bounded disturbances on the boundary can be expected to spread into the interior of the domain. The results indicate that such behavior indeed can occur, but suggest an estimate of general form for the magnitudes of the solution and of its derivatives, analogous to classical bounds for harmonic functions. The qualitative behavior of the solutions we found displayed some striking and unexpected features. As a corollary of the study, we obtain two new examples of non-uniqueness for stationary solutions at large Reynolds numbers.

Series: PDE Seminar

The usual boundary condition adjoined to a second order elliptic equation is the Dirichlet problem, which prescribes the values of the solution on the boundary. In many applications, this is not the natural boundary condition. Instead, the value of some directional derivative is given at each point of the boundary. Such problems are usually considered a minor variation of the Dirichlet condition, but this talk will show that this problem has a life of its own. For example, if the direction changes continuously, then it is possible for the solution to be continuously differentiable up to a merely Lipschitz boundary. In addition, it's possible to get smooth solutions when the direction changes discontinuously as well.

Series: PDE Seminar

In this talk we will consider three different numerical methods for solving nonlinear PDEs:

- A class of Godunov-type second order schemes for nonlinear conservation laws, starting from the Nessyahu-Tadmor scheme;
- A class of L1 -based minimization methods for solving linear transport equations and stationary Hamilton- Jacobi equations;
- Entropy-viscosity methods for nonlinear conservation laws.

All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.

Series: PDE Seminar

We consider a system of hyperbolic-parabolic equations describing the material instability mechanism associated to the formation of shear bands at high strain-rate plastic deformations of metals. Systematic numerical runs are performed that shed light on the behavior of this system on various parameter regimes. We consider then the case of adiabatic shearing and derive a quantitative criterion for the onset of instability: Using ideas from the theory of relaxation systems we derive equations that describe the effective behavior of the system. The effective equation turns out to be a forward-backward parabolic equation regularized by fourth order term (joint work with Th. Katsaounis and Th. Baxevanis, Univ. of Crete).