Date | Speaker | Title & Abstract |
April 18 |
Tarek Elgindi (Princeton) |
Title: On Singular Vortex Patches Abstract: Since the seminal work of Yudovich in 1963, it has been known that for a given uniformly bounded and compactly supported initial vorticity profile, there exists a unique global solution to the 2d incompressible Euler equation. A special class of Yudovich solutions are so-called vortex patch solutions where the vorticity profile is the characteristic function of an (evolving) bounded set in $\mathbb{R}^2.$ In 1993 Chemin and Bertozzi-Constantin proved that sufficiently high regularity of the boundary is propagated for all time. Since then, there have been numerous numerical and rigorous works on understanding the long-time dynamics of smooth vortex patches as well as the short time dynamics of vortex patches with corners. In this work, we consider two regimes; one where we prove well-posedness and the other where we prove ill-posedness. First, for vortex patches with corners enjoying a certain symmetry property at the corners, we prove global propagation of the corners; we also give examples where these vortex patches cusp in infinite time. Second, we prove that vortex patches with a single corner (which do not satisfy the symmetry condition) immediately cease to have a corner. This is joint work with I. Jeong. |
April 25 |
Tau Shean Lim (UWisc) |
Title: Propagation of Reactions in Levy Diffusions Abstract: We study reaction-diffusion equations u_{t} = L u + f(u) with homogeneous reactions f and diffusion operators L arising from the theory of Levy processes, with emphasis on propagation phenomena. The classical diffusion case (L = Laplacian) has been well-studied, including questions about traveling fronts, wavefront propagation, existence of spreading speeds, etc. After a brief review of the one-dimensional theory, we will concentrate on the case of nonlocal diffusions in several dimensions. We will discuss questions concerning long time dynamics of solutions, including spreading vs. quenching and existence of spreading speeds. |
May 2 |
Hamid Hezari (UCI) |
Title: Inverse spectral problems for strictly convex domains Abstract: I will talk about a result that is motivated by the work of De Simoi-Kaloshin-Wei. It concerns inverse spectral problems for strictly convex domains with one reflectional symmetry. The two key ingredients are wave trace formulas of Guillemin-Melrose and asymptotic of periodic billiard orbits of rotation numbers 1/q. |
May 16 |
Mihai Tohaneanu (U Kentucky) |
Title: Global existence for quasilinear wave equations close to
Schwarzschild. Abstract: We study the quasilinear wave equation ▢_{g}u=0, where the metric g depends on u and equals the Schwarzschild metric when u is identically 0. Under a couple of assumptions on the metric g near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad. |
June 6 |
Victor Lie (Purdue) |
Title: The pointwise convergence of Fourier Series near
L^{1}. Abstract: In our talk we will discuss the old and celebrated question regarding the pointwise behavior of Fourier Series near L^{1}. This presentation will include * The resolution of Konyagin's conjecture (ICM, Madrid 2006) on the pointwise convergence of Fourier Series along lacunary subsequences; * The L^{1}-strong convergence of Fourier Series along lacunary subsequences. * Recent progress on the L^{1}-strong convergence of (the full) Fourier Series. We end with several considerations on the relevance/impact of the above items on the subject of the pointwise convergence of Fourier Series. |
June 8 Note: Special Date |
Vera Hur (UIUC) |
Title: Full-dispersion shallow water models and the Benjamin-Feir
instability. Abstract: In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights. I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois). |
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 |
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). |