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
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||
Special day (Thursday) and location:
B139 Van Vleck
|Gui-Qiang Chen (Oxford)|| Supersonic Flow onto Solid Wedges,
Multidimensional Shock Waves and Free Boundary Problems
|April 3||Zhenfu Wang (Maryland)||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|
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.
We consider the patch evolution under the 2D Euler dynamics and study how the geometry of the boundary can deteriorate in time.
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.
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.
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.
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.
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.
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.
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.
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.