Date | Speaker | Title & Abstract |
September 29 11:00a-11:50a in AP&M 7421 Note: Special date. |
Tau Shean Lim (UWisc) |
Title: Traveling Fronts for Reaction-Diffusion Equations with Ignition Reactions and Levy Diffusion
Operators Abstract: We discuss traveling front solutions u(t,x) = U(x-ct) of reaction-diffusion equations u_t = Lu + f(u) in 1d with ignition reactions f and diffusion operators L generated by symmetric Levy processes X_t. Existence and uniqueness of fronts are well-known in the cases of classical diffusion (i.e., when L is the Laplacian) as well as some non-local diffusion operators. We extend these results to general Levy operators, showing that a weak diffusivity in the underlying process - in the sense that the first moment of X_1 is finite - gives rise to a unique (up to translation) traveling front. We also prove that our result is sharp, showing that no traveling front exists when the first moment of X_1 is infinite. |
October 11 | Zaher Hani (GTech) |
Title: Effective dynamics of nonlinear Schrodinger equations on large domains. Abstract: While the long-time behavior of small amplitude solutions to nonlinear dispersive and wave equations on Euclidean spaces (R^n) is relatively well-understood, the situation is marked different on bounded domains. Due to the absence of dispersive decay, a very rich set of dynamics can be witnessed even starting from arbitrary small initial data. In particular, the dynamics in this setting is characterized by out-of-equilibrium behavior, in the sense that solutions typically do not exhibit long-time stability near equilibrium configurations. This is even true for equations posed on very large domains (e.g. water waves on the ocean) where the equation exhibits very different behaviors at various time-scales. In this talk, we shall consider the nonlinear Schr\"odinger equation posed on a large box of size $L$. We will analyze the various dynamics exhibited by this equation when $L$ is very large, and exhibit a new type of dynamics that appears at a particular large time scale (that we call the resonant time scale). The rigorous derivation of this dynamics relies heavily on tools from analytic number theory. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah (all at Courant Institute, NYU). |
Date | Speaker | Title & Abstract |
March 14 |
Jessica Lin (UWisc) |
Title: Stochastic Homogenization for Reaction-Diffusion Equations Abstract: One way of modeling phenomena in ''typical" physical settings is to study PDEs in random environments. The subject of stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to PDEs with random coefficients. Specifically, we are interested in the following: if the random effects are microscopic compared to the length scale at which we observe the phenomena, can we predict the behavior which takes place on average? For certain models of PDEs and under suitable hypotheses on the environment, the answer is affirmative. In this talk, I will focus on the stochastic homogenization for reaction-diffusion equations with both KPP and ignition nonlinearities. In the large-scale-large-time limit, the behavior of typical solutions is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. In particular, we prove the existence of deterministic asymptotic speeds of propagation for reaction-diffusion equations in random media with both compactly supported and front-like initial data. Such models are relevant for predicting the evolution of a population or the spread of a fire in a heterogeneous environment. This talk is based on joint work with Andrej Zlatos. |
March 21 |
Jason Metcalfe (UNC) |
Title: Local well-posedness for quasilinear Schrodinger equations Abstract: I will speak on a recent joint study with J. Marzuola and D. Tataru which proves low regularity local well-posedness for quasilinear Schroedinger equations. Similar results were previously proved by Kenig, Ponce, and Vega in much higher regularity spaces using an artificial viscosity method. Our techniques, and in particular the spaces in which we work, are motivated by those used by Bejenaru and Tataru for semilinear equations. |
Date | Speaker | Title & Abstract |
May 16 |
Mihai Tohaneanu (U Kentucky) |
Title: TBA Abstract: TBA |
June 6 |
Victor Lie (Purdue) |
Title: TBA Abstract: TBA |
June 8 |
Vera Hur (UCI) |
Title: TBA Abstract: TBA |