Difference between revisions of "PDE Geometric Analysis seminar"
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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 timedependent 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.  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 timedependent 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.  
−  ===  +  ===Jeffrey Streets=== 
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 KahlerRicci flow to this setting. I will formulate a natural CalabiYau type conjecture based on Hitchin/Gualtieri's definition of generalized CalabiYau 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.  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 KahlerRicci flow to this setting. I will formulate a natural CalabiYau type conjecture based on Hitchin/Gualtieri's definition of generalized CalabiYau 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. 
Revision as of 13:29, 5 April 2017
The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm  4:30pm, unless indicated otherwise.
Contents
Previous PDE/GA seminars
Tentative schedule for Fall 2017
PDE GA Seminar Schedule Spring 2017
date  speaker  title  host(s) 

January 23 Special time and location: 33:50pm, 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 almostconvex cones  Kim & Tran 
February 27  Ben Seeger (University of Chicago)  Homogenization of pathwise HamiltonJacobi 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 ShigesadaKawasakiTeramoto system  Smith 
March 13  Sona Akopian (UTAustin)  Global $L^p$ well posedness of the Boltzmann equation with an anglepotential concentrated collision kernel.  Kim 
March 27  Analysis/PDE seminar  Sylvia Serfaty (Courant)  MeanField Limits for GinzburgLandau vortices  Tran 
March 29  Wasow lecture  Sylvia Serfaty (Courant)  Microscopic description of Coulombtype systems 

March 30 Special day (Thursday) and location: B139 Van Vleck 
GuiQiang 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  SiaoHao 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 (UCIrvine)  Generalized Kahler Ricci flow and a generalized Calabi conjecture  Bing Wang 
Abstracts
Sigurd Angenent
The HuiskenHamiltonGage 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 illposed 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 "almostconvex" 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 HamiltonJacobi equations with "rough" multiplicative driving signals. In doing so, I derive a new wellposedness result when the Hamiltonian is smooth, convex, and positively homogenous. I also demonstrate that equations involving multiple driving signals may homogenize or exhibit blowup.
Sona Akopian
Global $L^p$ well posedness of the Boltzmann equation with an anglepotential concentrated collision kernel.
We solve the Cauchy problem associated to an epsilonparameter 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 pseudoMaxwell 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
MeanField Limits for GinzburgLandau vortices
GinzburgLandau type equations are models for superconductivity, superfluidity, BoseEinstein condensation. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complexvalued 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 GinzburgLandau equation or the GrossPitaevskii (=Schrodinger GinzburgLandau) equation.
GuiQiang 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 straightsided 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 elliptichyperbolic 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 Nparticles distribution and the expected limit which solves the corresponding McKeanVlasov PDE. This implies the Mean Field limit to the McKeanVlasov 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 threedimensional fluid flow on the torus that also keeps track of the local temperature. The momentum equation is the same as for NavierStokes, 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 LadyzhenskayaProdiSerrin criterion for ordinary NavierStokes) on the thermally weighted enstrophy for classical solutions to the coupled system.
Siaohao 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 timedependent 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.
Jeffrey Streets
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 KahlerRicci flow to this setting. I will formulate a natural CalabiYau type conjecture based on Hitchin/Gualtieri's definition of generalized CalabiYau 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.