Spring 2019

Organizer : Dr. Yongki Lee and Dr. Yuanzhen Shao 

• Math Physics Seminar

• Joint talk with Department Colloquium 

  • Date: Friday, March 1 

  • Time: 3:30 – 4:30 PM 

  • Location: Math/Physics Bldg 3314 

  • Speaker: Dr. Patrick Guidotti (University of California Irvine) 

  • Title: A Novel Optimization Approach to Fictitious Domain Methods

  • Abstract: There are many ways to discretize genereal initial boundary value problems of mathematical physics. So-called embedding methods attempt at extending equations in complex domains to a larger and simpler domain, where they admit simpler discretizations. Due the singularities caused by the extension, however, the methods have limited accuracy or require computational intensive smooth extensions of the data. We propose a new methodology which allows one to circumvent these difficulties, admits straight-forward discretizations of tunable and high accuracy, which will be proven effective for problems in general geometries, for non-constant coefficients and general boundary conditions in dimensions 2 and 3.

  • Date: Thursday, March 28

  • Time: 3:00 – 4:00 PM 

  • Location: MP1315

  • Speaker: Dr. Yuanzhen Shao (Georgia Southern) 

  • Title:  The Fractional Porous Medium Equation on Manifolds with Conic Singularities 

  • Abstract: Due to the need to model long range diffusive interaction, during the last decade there has been a growing interest in considering diffusion equations involving non-local operators, e.g. the fractional powers of differential operators. In this talk, I will report some recent work with Nikolaos Roidos on the fractional porous medium equation on manifolds with cone-like singularities. I will show that most of the properties of the usual (local) porous medium equation, like existence, uniqueness of weak solution, comparison principle, conservation of mass, are inherited by the non-local version.

• Joint talk with Department Colloquium 

  • Date: Friday, April 5 

  • Time: 3:30 – 4:30 PM 

  • Location: Math/Physics Bldg 3314 

  • Speaker: Dr. Kai Yang (FIU) 

  • Title: Singularity formations in the nonlinear Schrödinger-type equations.

  • Abstract: We present two different approaches that showing the stable blow-up solutions in the L^2-critical NLS equation (the log-log blow-up dynamics up to the dimension 12). One approach is the direct numerical simulation via the "Dynamic rescaling method", and then combined with the asymptotic analysis of the rescaled equation. The other approach is to prove the Spectral property with the numerical assistance. Then, use such Spectral property in the analysis to complete the proof. Then, we would like to show the similar numerical results (from the Dynamic rescaling method) that we obtained for the generalized Hartree equation (the NLS equation with non-local potential).

  • Date: Thursday, April 11

  • Time: 3:00 – 4:00 PM 

  • Location: MP1315

  • Speaker: Dr. Yongki Lee (Georgia Southern) 

  • Title:   Wave breaking conditions for the traffic flow models and the Whitham-type equation .   

  • Abstract:  In this talk, we discuss threshold conditions for wave breaking in a class of non-local conservation laws with concavity changing flux. From a class of non-local conservation laws, the Riccati-type ODE system that governs a solution’s gradient is obtained. The changes in concavity of the flux correspond to the sign changes in the leading coefficient functions of the ODE system. We identify the blow-up condition of this structurally generalized Riccati-type ODE. The method is illustrated vis the traffic flow models with nonlocal-concave-convex flux and the Whitham-type equation with nonlinear drift.

• Joint talk with Department Colloquium 

  • Date: Friday, April 26

  • Time: 3:30 – 4:30 PM 

  • Location: Math/Physics Bldg 3314 

  • Speaker: Dr. Chenchen Mou (University of California Los Angeles) 

  • Title: Weak Solutions of Mean Field Game Master Equations

  • Abstract: In this talk we study master equations arising from mean field game problems, under the crucial monotonicity conditions. Classical solutions of such equations require very strong technical conditions. Moreover, unlike the master equations arising from mean field control problems, the mean field game master equations are non-local and even classical solutions typically do not satisfy the comparison principle, so the standard viscosity solution approach seems infeasible. We shall propose a notion of weak solution for such equations and establish its wellposedness. Our approach relies on a new smooth mollifier for functions of measures, which unfortunately does not keep the monotonicity property, and the stability result of master equations. The talk is based on a joint work with Jianfeng Zhang.

  • Date: Thursday, May 2

  • Time: 3:00 – 4:00 PM 

  • Location: MP1315

  • Speaker: Dr. Shijun Zheng (Georgia Southern) 

  • Title:  Orbital Stability of Higher-Order Hamiltonians

  • Abstract: I will address the stability problem for a class of dispersive Hamiltonian systems that model nonlinear quantum waves. In particular, I will consider our recent joint work on the bi-harmonic NLSE with a second-order perturbation. I will elaborate the stability theory as well as the threshold dynamics and blowup behavior involving the celebrated log-log law for the wave collapse. The variational method we apply will be a profile decomposition, which dates back to P.L. Lions, and more recently, Gerard et al. The profile decomposition plus the spectral calculus are efficient tool that can allow us to treat other systems like the magnetic NLSE, fractional NLSE with an harmonic potential, and Dirac equations.