PDE Geometric Analysis seminar

The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.

Abstracts

Sigurd Angenent

The Huisken-Hamilton-Gage theorem on compact convex solutions to MCF shows that in forward time all solutions do the same thing, namely, they shrink to a point and become round as they do so. Even though MCF is ill-posed in backward time there do exist solutions that are defined for all t<0 , and one can try to classify all such “Ancient Solutions.” In doing so one finds that there is interesting dynamics associated to ancient solutions. I will discuss what is currently known about these solutions. Some of the talk is based on joint work with Sesum and Daskalopoulos.

Serguei Denissov

We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.

Bing Wang

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R3. This is a joint work with H.Z. Li.

Eric Baer

We discuss a recent result showing that a characterization of isoperimetric sets (that is, sets minimizing a relative perimeter functional with respect to a fixed volume constraint) inside convex cones as sections of balls centered at the origin (originally due to P.L. Lions and F. Pacella) remains valid for a class of "almost-convex" cones. Key tools include compactness arguments and the use of classically known sharp characterizations of lower bounds for the first nonzero Neumann eigenvalue associated to (geodesically) convex domains in the hemisphere. The work we describe is joint with A. Figalli.

Ben Seeger

I present a homogenization result for pathwise Hamilton-Jacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new well-posedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blow-up.

Sona Akopian

Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel.

We solve the Cauchy problem associated to an epsilon-parameter family of homogeneous Boltzmann equations for very soft and Coulomb potentials. Proposed in 2013 by Bobylev and Potapenko, the collision kernel that we use is a Dirac mass concentrated at very small angles and relative speeds. The main advantage of such a kernel is that it does not separate its variables (relative speed $u$ and scattering angle $\theta$) and can be viewed as a pseudo-Maxwell molecule collision kernel, which allows for the splitting of the Boltzmann collision operator into its gain and loss terms. Global estimates on the gain term gives us an existence theory for $L^1_k \capL^p$ with any $k\geq 2$ and $p\geq 1.$ Furthermore the bounds we obtain are independent of the epsilon parameter, which allows for analysis of the solutions in the grazing collisions limit, i.e., when epsilon approaches zero and the Boltzmann equation becomes the Landau equation.

Sylvia Serfaty

Mean-Field Limits for Ginzburg-Landau vortices

Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. This talk will review some results on the derivation of effective models to describe the statics and dynamics of these vortices, with particular attention to the situation where the number of vortices blows up with the parameters of the problem. In particular we will present new results on the derivation of mean field limits for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.

Gui-Qiang Chen

Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems

When an upstream steady uniform supersonic flow, governed by the Euler equations, impinges onto a symmetric straight-sided wedge, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -- the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy conditions. The fundamental issue -- whether one or both of the steady weak and strong shocks are physically admissible solutions -- has been vigorously debated over the past eight decades. In this talk, we discuss some of the most recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. The corresponding stability problems can be formulated as free boundary problems for nonlinear partial differential equations of mixed elliptic-hyperbolic type, whose solutions are fundamental for multidimensional hyperbolic conservation laws. Some further developments, open problems, and mathematical challenges in this direction are also addressed.

Zhenfu Wang

Title: Mean field limit for stochastic particle systems with singular forces

Abstract: We consider large systems of particles interacting through rough interaction kernels. We are able to control the relative entropy between the N-particles distribution and the expected limit which solves the corresponding McKean-Vlasov PDE. This implies the Mean Field limit to the McKean-Vlasov system together with Propagation of Chaos through the strong convergence of all the marginals. The method works at the level of the Liouville equation and relies on precise combinatorics results.

Andrei Tarfulea

We consider a model for three-dimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for Navier-Stokes, however the kinematic viscosity grows as a function of the local temperature. The temperature is, in turn, fed by the local dissipation of kinetic energy. Intuitively, this leads to a mechanism whereby turbulent regions increase their local viscosity and dissipate faster. We prove a strong a priori bound (that would fall within the Ladyzhenskaya-Prodi-Serrin criterion for ordinary Navier-Stokes) on the thermally weighted enstrophy for classical solutions to the coupled system.

Siao-hao Guo

Analysis of Velázquez's solution to the mean curvature flow with a type II singularity

Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.

Jianfeng Lu

Evolution of crystal surfaces: from mesoscopic to continuum models

In this talk, we will discuss some of our recent results on understanding various models for crystal surface evolution at different physical scales; in particular, we will focus on the connection of mesoscopic and continuum (PDE) models for crystal surface relaxation and also discuss several PDEs arising from different physical scenarios. Many interesting open problems remain to be studied. Based on joint work with Yuan Gao, Jian-Guo Liu, Dio Margetis and Jeremy Marzuola.

Chris Henderson

A local-in-time Harnack inequality and applications to reaction-diffusion equations

The classical Harnack inequality requires one to look back in time to obtain a uniform lower bound on the solution to a parabolic equation. In this talk, I will introduce a Harnack-type inequality that allows us to remove this restriction at the expense of a slightly weaker bound. I will then discuss applications of this bound to (time permitting) three non-local reaction-diffusion equations arising in biology. In particular, in each case, this inequality allows us to show that solutions to these equations, which do not enjoy a maximum principle, may be compared with solutions to a related local equation, which does enjoy a maximum principle. Precise estimates of the propagation speed follow from this.

Jeffrey Streets

Generalized Kahler Ricci flow and a generalized Calabi conjecture

Generalized Kahler geometry is a natural extension of Kahler geometry with roots in mathematical physics, and is a particularly rich instance of Hitchin's program of `generalized geometries.' In this talk I will discuss an extension of Kahler-Ricci flow to this setting. I will formulate a natural Calabi-Yau type conjecture based on Hitchin/Gualtieri's definition of generalized Calabi-Yau equations, then introduce the flow as a tool for resolving this. The main result is a global existence and convergence result for the flow which yields a partial resolution of this conjecture, and which classifies generalized Kahler structures on hyperKahler backgrounds.

date	speaker	title	host(s)
September 11	Mihaela Ifrim (UW)	TBD	Kim & Tran
September 18	Longjie Zhang (University of Tokyo)	TBD	Angenent
September 22, VV B239 4:00pm	Jaeyoung Byeon (KAIST)	Colloquium:	Rabinowitz
September 25	Tuoc Phan (UTK)	TBD	Tran
September 26, VV B139 4:00pm	Hiroyoshi Mitake (Hiroshima University)	Joint Analysis/PDE seminar	Tran
September 29, VV901 2:25pm	Dongnam Ko (CMU & SNU)	a joint seminar with ACMS: TBD	Shi Jin & Kim
October 2	No seminar due to a KI-Net conference
October 9	Sameer Iyer (Brown University)	TBD	Kim
October 16	Jingrui Cheng (UW)	TBD	Kim & Tran
October 23	Donghyun Lee (UW)	TBD	Kim & Tran
November 6	Jingchen Hu (USTC and UW)	TBD	Kim & Tran