# 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.

# PDE GA Seminar Schedule Spring 2017

date speaker title host(s)
January 23
Special time and location:
3-3:50pm, B325 Van Vleck
Sigurd Angenent (UW) Ancient convex solutions to Mean Curvature Flow Kim & Tran
January 30 Serguei Denissov (UW) Instability in 2D Euler equation of incompressible inviscid fluid Kim & Tran
February 6 - Wasow lecture Benoit Perthame (University of Paris VI) Jin
February 13 Bing Wang (UW) The extension problem of the mean curvature flow Kim & Tran
February 20 Eric Baer (UW) Isoperimetric sets inside almost-convex cones Kim & Tran
February 27 Ben Seeger (University of Chicago) Homogenization of pathwise Hamilton-Jacobi equations Tran
March 7 - Mathematics Department Distinguished Lecture Roger Temam (Indiana University) On the mathematical modeling of the humid atmosphere Smith
March 8 - Analysis/Applied math/PDE seminar Roger Temam (Indiana University) Weak solutions of the Shigesada-Kawasaki-Teramoto system Smith
March 13 Sona Akopian (UT-Austin) Global $L^p$ well posed-ness of the Boltzmann equation with an angle-potential concentrated collision kernel. Kim
March 27 - Analysis/PDE seminar Sylvia Serfaty (Courant) Mean-Field Limits for Ginzburg-Landau vortices Tran
March 29 - Wasow lecture Sylvia Serfaty (Courant) Microscopic description of Coulomb-type systems

March 30
Special day (Thursday) and location:
B139 Van Vleck
Gui-Qiang Chen (Oxford) Supersonic Flow onto Solid Wedges, Feldman

April 3 Zhenfu Wang (Maryland) Mean field limit for stochastic particle systems with singular forces Kim
April 10 Andrei Tarfulea (Chicago) Improved estimates for thermal fluid equations Baer
April 17 Siao-Hao Guo (Rutgers) Analysis of Velázquez's solution to the mean curvature flow with a type II singularity Lu Wang

April 24 Jianfeng Lu TBA Li
April 25- joint Analysis/PDE seminar Chris Henderson (Chicago) TBA Lin
May 1st Jeffrey Streets (UC-Irvine) Bing Wang

# 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.