## Department of Mathematics, Purdue University PDE Seminar

Fall 2020

Talks in Fall 2020 will take place virtually at 3:30-4:30pm, Thursday. If you have any questions about a particular seminar, please contact the organizer at: wang2482@purdue.edu.
The Zoom meeting link is: Meeting ID: 98551029625; Meeting Passcode: 570121

Date SPEAKER and TITLE Host
September 10 Changyou Wang, Purdue University
TITLE: On phase transition problem between isotropic and nematic states of liquid crystals
ABSTRACT : In this talk, I will discuss the phase transition phenomena between the isotropic and nematic states within the framework of Ericksen theory of liquid crystals with variable degrees of orientations. Treating it as the singular perturbation problems with in the Gamma convergence theory, we will show that the sharp interface formed between isotropic and nematic states is an area minimizing surface. Under suitable assumptions either on the strong anchoring boundary values on the boundary of a bounded domain or the volume constraint of nematic regions in the entire space, we also show that the limiting nematic liquid configuration in the nematic region is a minimizer of the corresponding Oseen-Frank energy with either homeotropic or planar anchorings on the free sharp interface pending on the relative sizes of leading Frank elasticity coefficients. This is a joint work with Fanghua Lin.
Wang
September 17 Jiajun Tong, UCLA
TITLE: Interface Dynamics in a Two-phase Tumor Growth Model
ABSTRACT: We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. Our focus will be on the dynamics of two moving interfaces in the model that separate the tumor, the normal tissue, and the exterior vacuum. We will show local-in-time existence and uniqueness of strong solutions for their evolution starting from a nearly radial initial configuration, assuming that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering. We will briefly discuss possible ill-posedness if these conditions are violated. This is joint work with Inwon Kim
Wang
September 24 Hengrong Du, Purdue University
TITLE: Weak compactness of nematic liquid crystal flows in 2D
ABSTRACT : In this talk, I will discuss the weak compactness property of solutions to the Ericksen--Leslie system modeling the hydrodynamics of nematic liquid crystals in 2D. The system can be viewed as a coupling between the Navier--Stokes equations for velocity fields and the heat flows of harmonic maps for director fields via a highly nonlinear quadratic term called the Ericksen stress tensor. We show that a kind of "concentration cancellation" occurs in weak solution sequences to the Ericksen--Leslie system and its Ginzburg--Landau type approximation. Roughly speaking, it says that the Ericksen stress terms are insensible to concentrations of director energy so that we can pass the weak limit despite the appearance of singularities. The proof is based on the $L^p$-estimate of the Hopf differential and the Pohozaev type argument. This is a joint work with Tao Huang(Wayne State) and Changyou Wang(Purdue).
Wang
October 1 Peter Sternberg, Indiana University, Bloomington
TITLE: A One-Dimensional Variational Problem for Cholesteric Liquid Crystals with Disparate Elastic Constants
ABSTRACT : We consider a one-dimensional variational problem arising in connection with a model for cholesteric liquid crystals. The principal feature of our study is the assumption that the twist deformation of the nematic director incurs much higher energy penalty than other modes of deformation. The appropriate ratio of the elastic constants then gives a small parameter epsilon entering an Allen-Cahn-type energy functional augmented by a twist term. We consider the behavior of the energy as epsilon tends to zero. We demonstrate existence of local energy minimizers classified by their overall twist, find the Gamma-limit of these energies and show that it consists of twist and jump terms. This is joint work with Dmitry Golovaty (Akron) and Michael Novack (UT Austin).
Wang
October 8 Yanick Sire, Johns Hopkins University
TITLE:New results on constant Q-curvature metrics
ABSTRACT: In the recent years, several major results have been obtained in the problem of finding a constant Q-curvature metric in a given conformal class in dimensions bigger than 5. This talk will cover new results concerning existence and multiplicity of such metrics. I will first present a rather general geometric approach to prove existence and multiplicity of regular metrics, giving several explicit examples. Then I will move to the case of singular metrics, i.e. complete metrics with constant Q-curvature outside of a closed set. This requires to develop several tools to handle 4th order equations (but applicable actually to higher order ones). I will also provide some explicit examples of such metrics and investigate their multiplicity. I will state open problems as well.
Wang
October 15 John Lewis, University of Kentucky
TITLE: On a Theorem of Wolff Revisited
ABSTRACT : Fatou's theorem for harmonic functions states that if u is a bounded harmonic function in the unit ball or upper half space in R^n, then u has finite radial limits almost everywhere. Tom Wolff proved in 1983 that this theorem fails for solutions to p-Laplace equation in the upper half space of R^2 when p > 2. A few years later I used a conjugate function argument" to show his theorem generalizes to p>1. In our paper (to appear Jour. Anal. Math.) with the same name as the above title, we extend Wolff's theorem for p>1 to the unit disk in R^2. In this talk we first give some preliminary material on harmonic and p-harmonic functions. After that we discuss some of the brilliant ideas Wolff used in his construction. Finally we outline our proof. Time permitting we will also discuss two possible definitions of p harmonic measure and discuss our work in the above paper concerning generalizations of relatively recent work of Llorente, Manfredi, and Wu (on one definition of p harmonic measure in the upper half space) to the unit disk in R^2.
Wang
October 22 Aaron Yip, Purdue University
TITLE: A revisit of homogenization of motion by mean curvature
ABSTRACT: Despite a long history and many results on motion by mean curvature and homogenization, there are still many open questions when these two problems are put together. This is a typical question of gradient flow on wiggly energy landscapes. This talk will revisit the problem of motion by mean curvature in heterogeneous medium. We will attempt this from a purely energetic perspective. We take the point of view of convergence of gradient flow in the framework of Sandier-Serfaty together with the recent development and extension by Mielke and co-authors. This is joint work in progress of T. Laux.
Wang
October 29 Hung Tran, University of Wisconsin, Madison (Cancelled)
TITLE: tba
ABSTRACT : tba
Wang
November 5 Emanuel Indrei, Purdue University
TITLE: The equilibrium shape of a planar crystal in a convex coercive background is convex
ABSTRACT: I'll discuss my recent solution to a long-standing open problem posed by Almgren: https://arxiv.org/abs/2008.02238
Wang
November 12 Alaa Haj Ali, Purdue University
TITLE: A boundary obstacle problem for the bi-Laplace operator
ABSTRACT :We study the boundary obstacle problem for the bi-Laplacian in the upper unit ball with boundary condition $(\Delta u)_{y}=\lambda_- (u^-)^{p-1}-\lambda_+ (u^+)^{p-1}$ on the flat part $\Gamma$ of the boundary. Here $\lambda_{\pm}$ are positive constants and $u^{\pm}$ are the positive and negative parts of $u$. We establish the well-posedness of the problem and we study the optimal regularity of a solution. Then we prove some growth rate and non-degeneracy results of the solution at points on the free boundary $\Gamma \cap (\partial\{u>0\}\cup \partial\{u<0\})$ and we derive some properties related to the structure of the singular set. This is a joint work with Donatella Danielli.
Wang
November 19 Yuanzhen Shao, University of Alabama
TITLE: Large Time Behavior of the Fractional Porous Medium Equation on Riemannian Manifolds via Fractional Logarithmic Sobolev Inequalities
ABSTRACT : In 1995, Carlen and Loss proposed a novel approach to study the asymptotics the 2–D Navier–Stokes equation by using a Logarithmic Sobolev inequality. Later, this idea was adopted by Bonforte and Grillo to study the asymptotics of the Porous Medium Equation. In this talk, I will discuss how to derive various Sobolev type inequality involving fractional Laplacian. Then I will use these inequalities to study the smooth effect and asymptotic behavior of the Fractional Porous Medium Equation on complete Riemannian manifold
Wang