TITLE:  Scattering Resonances on Hyperbolic Manifolds as a Model of Chaotic Scattering

ABSTRACT:  Scattering resonances represent "almost standing waves" in a scattering system which have a nite lifetime as measured by energy decay in a nte region.  In this survey talk well review the basics of scattering theory on geometrically nite, real hyperbolic manifolds and their role as models of open chaotic systems. As such they have attracted the interest of both mathematicians and physicists. Work to be discussed includes the work of Patterson and Perry and papers by Borthwick, Guillope-Zworski, Guillarmou, Naud, and others.  Real hyperbolic manifolds provide a useful "laboratory" for scattering because their symmetries allow for the use of powerful methods from the theory of automorphic functions, dynamics, and the theory of Fuchsian groups. We'll discuss the connection between scattering resonances and Helberg's zeta function for a hyperbolic surface, and in turn the connection between Selberg's zeta function and the Ruelle zeta function from dynamical systems. Through this connection one can uncover close relationships between the Hausdor  dimension of the trapped set for geodesic ow on the one hand, and the distribution of scattering resonances on the other.

Title:  Gradient estimates for the Stokes semigroup subject Neumann boundary conditions in bounded convex domains.

Abstract available at http://www.ms.uky.edu/~hislop/PDEseminarS2015

Abstract available at: 

http://www.ms.uky.edu/~rbrown/misc/perez-2015-03-03.pdf

Title:  Fluid PDEs with partial or fractional dissipation.

Title:  Boundary integral equations of coupled thermoelastodynamics

Title:  Using the method of layer potentials to solve a mixed boundary value problem

Abstract:  Following the exposition by William McLean in his book Strongly Elliptic Systems and Boundary Integral Equations, we use the method of layer potentials to show that on a bounded Lipschitz domain, the mixed problem for Laplace’s equation is equivalent to a 2 × 2 system of boundary integral equations.

Title: Self-improvement properties for nonlocal equations

Abstract:  I will present some results related to generalization of Meyers result to nonlocal equation. It happens that any weak solution of a nonlocal equation with data in L2 is automatically better at the integrability AND differentiability scale. This is a completely new phenomenon relying on the nonlocality of the operator. The proof is based on a new stopping time argument and a suitable generalization of Gehring lemma.

Title:  Augmented eigenfucntions: a new spectral object appearing in the integral representation of the solution of linear initial-boundary value problems.

Abstract:  We study initial-boundary value problems for linear, constant-coefficient partial differential equations of arbitrary order, on a finite or semi-infinite domain, with arbitrary boundary conditions. It has been shown that the recent Unified Transform Method of Fokas can be used to solve all such classically well-posed problems. The solution thus obtained is expressed as an integral, which represents a new kind of spectral transform. We compare the new method, and its solution representation, with classical Fourier transform techniques, and the resulting solution representation. In doing so, we discover a new species of spectral object, encoded by the spectral transforms of the Unified Method.

Title:  Extremal functions in modules of systems of measures

Abstract:  We study Fuglede’s p-modules of systems of measures in condensers in the Euclidean spaces. First, we generalize the result by Rodin that provides a way to compute the extremal function and the 2-module of a family of curves in the plane to a variety of other settings. More specifically, in the Euclidean space we compute the p-module of images of families of connecting curves and families of separating sets with respect to the plates of a condenser under homeomorphisms with some assumed regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin’s theorem to the spherical ring domain on the Heisenberg group. Applications to special functions and examples will be provided. Joint work with Melkana Brakalova and Irina Markina.

Title:  On a generalized Derivative Nonlinear SchrÖdinger equation

Abstract:  The Derivative Nonlinear SchrÖdinger equation (DNLS) equation iψt + ψxx + i |ψ|2 ψx = 0 is a canonical equation obtained from the Hall-MHD equations in a long-wave scaling, in the context of weakly nonlinear Alfvén waves propagating along an ambient magnetic field. It has the same scaling properties as the Nonlinear SchrÖdinger equation with quantic power law nonlinearity (L2-critical) that develop singularities in a finite time.  It also has the property of being completely integrable by the inverse scattering transform and has soliton solutions. In an effort to address the open question of long-time existence, we introduced recently an L2 -supercritical version of the DNLS equation by modifying the nonlinearity |ψ|2ψx to |ψ|2σψx (σ > 1). Numerical simulations indicate that a finite time singularity may occur, and provide a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile. The (complex valued) profile satisfies a nonlinear elliptic equation Qξξ −Q+ia(1/2σQ+ξQξ) − ibQξ + i |Q|2σQξ = 0, where the (real) coefficients a and b depend on σ (but not on the initial condition). Using methods of asymptotic analysis, we study the deformation of the functions Q, and parameters a, b as the nonlinearity σ tends to 1. We also check our analysis against a numerical integration of the profile equation with continuation type methods. This is an ongoing work with G. Simpson and Y. Cher.