## Seminars and Colloquia by Series

Series: PDE Seminar
Tuesday, November 27, 2018 - 15:00 , Location: skiles 006 , Yilun(Allen) Wu , The University of Oklahoma , , Organizer: Xukai Yan
A rotating star may be modeled as gas under self gravity with a given total mass and prescribed angular velocity. Mathematically this leads to the Euler-Poisson system. In this talk, we present an existence theorem for such stars that are rapidly rotating, depending continuously on the speed of rotation. No previous results using continuation methods allowed rapid rotation. The key tool for the result is global continuation theory via topological degree, combined with a delicate limiting process. The solutions form a connected set $\mathcal K$ in an appropriate function space. Take an equation of state of the form $p = \rho^\gamma$; $6/5 < \gamma < 2$, $\gamma\ne 4/3$. As the speed of rotation increases, we prove that either the density somewhere within the stars becomes unbounded, or the supports of the stars in $\mathcal K$ become unbounded. Moreover, the latter alternative must occur if $\frac43<\gamma<2$. This result is joint work with Walter Strauss.
Series: PDE Seminar
Tuesday, November 13, 2018 - 15:00 , Location: Skiles 005 , Prof. Shigeaki Koike , Tohoku University, Japan , Organizer: Ronghua Pan
We discuss bilateral obstacle problems for fully nonlinear second order uniformly elliptic partial differential equations (PDE for short) with merely continuous obstacles. Obstacle problems arise not only in minimization of energy functionals under restriction by obstacles but also stopping time problems in stochastic optimal control theory. When the main PDE part is of divergence type, huge amount of works have been done. However, less is known when it is of non-divergence type. Recently, Duque showed that the Holder continuity of viscosity solutions of bilateral obstacle problems, whose PDE part is of non-divergence type, and obstacles are supposed to be Holder continuous. Our purpose is to extend his result to enable us to apply a much wider class of PDE. This is a joint work with Shota Tateyama (Tohoku University).
Series: PDE Seminar
Tuesday, November 6, 2018 - 15:00 , Location: Skiles 006 , Hao Jia , University of Minnesota , , Organizer: Xukai Yan
The two dimensional Euler equation is globally wellposed, but the long time behavior of solutions is not well understood. Generically, it is conjectured that the vorticity, due to mixing, should weakly but not strongly converge as $t\to\infty$. In an important work, Bedrossian and Masmoudi studied the perturbative regime near Couette flow $(y,0)$ on an infinite cylinder, and proved small perturbation in the Gevrey space relaxes to a nearby shear flow. In this talk, we will explain a recent extension to the case of a finite cylinder (i.e. a periodic channel) with perturbations in a critical Gevrey space for this problem. The main interest of this extension is to consider the natural boundary effects, and to ensure that the Couette flow in our domain has finite energy. Joint work with Alex Ionescu.
Series: PDE Seminar
Tuesday, October 16, 2018 - 15:00 , Location: Skiles 006 , , Rutgers University , , Organizer: Ronghua Pan
We give derivative estimates for solutions to divergence form elliptic equations with piecewise smooth coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces.
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.