## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, November 13, 2018 - 15:00 , Location: Skiles 005 , Prof. Shigeaki Koike , Tohoku University, Japan , Organizer: Ronghua Pan
TBA
Series: PDE Seminar
Tuesday, October 16, 2018 - 15:00 , Location: Skiles 006 , , Rutgers University , , Organizer: Ronghua Pan
TBA
Series: PDE Seminar
Thursday, October 11, 2018 - 15:00 , Location: Skiles 005 , , African Institute for Mathematical Sciences, Tanzania , Organizer: Yao Yao
In this talk, we introduce several models of the so-called forward-forward Mean-Field Games (MFGs). The  forward-forward models arise in the study of  numerical schemes to approximate stationary MFGs. We establish a link between these models and a class of hyperbolic conservation laws. Furthermore, we investigate the existence of solutions and examine long-time limit properties. Joint work with Diogo Gomes and Levon Nurbekyan.
Series: PDE Seminar
Tuesday, September 25, 2018 - 15:00 , Location: Skiles 006 , , University of Chicago , , Organizer: Yao Yao
I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy.  Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion.  The talk is partially based on joint work with Charles Smart.
Series: PDE Seminar
Tuesday, September 4, 2018 - 15:00 , Location: Skiles 006 , , UCLA , Organizer: Yao Yao
We consider a class of nonlinear, degenerate drift-diffusion equations in R^d. By a scaling argument, it is widely believed that solutions are uniformly Holder continuous given L^p-bound on the drift vector field for p>d. We show the loss of such regularity in finite time for p≤d, by a series of examples with divergence free vector fields. We use a barriers argument.
Series: PDE Seminar
Tuesday, August 21, 2018 - 15:00 , Location: Skiles 006 , Professor Veli Shakhmurov , Okan University , , Organizer: Ronghua Pan
The boundary value and mixed value problems for linear and nonlinear degenerate abstract elliptic and parabolic equations are studied. Linear problems involve some parameters. The uniform L_{p}-separability properties of linear problems and the optimal regularity results for nonlinear problems are obtained. The equations include linear operators defined in Banach spaces, in which by choosing the spaces and operators we can obtain numerous classes of problems for singular degenerate differential equations which occur in a wide variety of physical systems.     In this talk, the classes of boundary value problems (BVPs) and mixed value problems (MVPs) for regular and singular degenerate differential operator equations (DOEs) are considered. The main objective of the present talk is to discuss the maximal regularity properties of the BVP for the degenerate abstract elliptic and parabolic equation     We prove that for f∈L_{p} the elliptic problem has a unique solution u∈ W_{p,α}² satisfying the uniform coercive estimate ∑_{k=1}ⁿ∑_{i=0}²|λ|^{1-(i/2)}‖((∂^{[i]}u)/(∂x_{k}^{i}))‖_{L_{p}(G;E)}+‖Au‖_{L_{p}(G;E)}≤C‖f‖_{L_{p}(G;E)} where L_{p}=L_{p}(G;E) denote E-valued Lebesque spaces for p∈(1,∞) and W_{p,α}² is an E-valued Sobolev-Lions type weighted space that to be defined later. We also prove that the differential operator generated by this elliptic problem is R-positive and also is a generator of an analytic semigroup in L_{p}. Then we show the L_{p}-well-posedness with p=(p, p₁) and uniform Strichartz type estimate for solution of MVP for the corresponding degenerate parabolic problem. This fact is used to obtain the existence and uniqueness of  maximal regular solution of the MVP for the nonlinear parabolic equation.
Series: PDE Seminar
Thursday, May 3, 2018 - 15:00 , Location: Skiles 005 , , Missouri University of Science and Technology , , Organizer:
The ground state solution to the nonlinear Schrödinger equation (NLS) is a global, non-scattering solution that often provides a threshold between scattering and blowup.  In this talk, we will discuss new, simplified proofs of scattering below the ground state threshold (joint with B. Dodson) in both the radial and non-radial settings.
Series: PDE Seminar
Thursday, April 26, 2018 - 15:00 , Location: Skiles 257 , , SISSA , , Organizer: Yao Yao
We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to  nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.
Series: PDE Seminar
Tuesday, April 24, 2018 - 15:00 , Location: Skiles 006 , , SISSA , , Organizer: Yao Yao
(Due to a flight cancellation, this talk will be moved to Thursday (Apr 26) 3pm at Skiles 257). We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to  nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.
Series: PDE Seminar
Tuesday, April 17, 2018 - 15:00 , Location: Skiles 006 , , Duke University , , Organizer: Yao Yao
Epitaxial growth is an important physical process for forming solid films or other nano-structures.  It occurs as atoms, deposited from above, adsorb and diffuse on a crystal surface.  Modeling the rates that atoms hop and break bonds leads in the continuum limit to degenerate 4th-order PDE that involve exponential nonlinearity and the p-Laplacian with p=1, for example.  We discuss a number of analytical results for such models, some of which involve subgradient dynamics for Radon measure solutions.