Difference between revisions of "PDE Geometric Analysis seminar"
(→Abstracts) 

Line 103:  Line 103:  
April 17  April 17  
 SiaoHao Guo (Rutgers)   SiaoHao Guo (Rutgers)  
−  [[# SiaoHao Guo   +  [[# SiaoHao Guo  Analysis of Velázquez's solution to the mean curvature flow with a type II singularity]] 
 Lu Wang   Lu Wang  
Line 153:  Line 153:  
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  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.  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. 
Revision as of 23:25, 8 March 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)  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  
April 3  Zhenfu Wang (Maryland)  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)  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.
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.
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.