PDE Geometric Analysis seminar: Difference between revisions

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=Abstracts=
=Abstracts=


===Tianling Jin===
=== ===
 
Holder gradient estimates for parabolic homogeneous p-Laplacian equations
 
We prove interior Holder estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous p-Laplacian equation
u_t=|\nabla u|^{2-p} div(|\nabla u|^{p-2}\nabla u),
where 1<p<\infty. This equation arises from tug-of-war like stochastic games with white noise. It can also be considered as the parabolic p-Laplacian equation in non divergence form. This is joint work with Luis Silvestre.
 
===Russell Schwab===
 
Neumann homogenization via integro-differential methods
 
In this talk I will describe how one can use integro-differential methods to attack some Neumann homogenization problems-- that is, describing the effective behavior of solutions to equations with highly oscillatory Neumann data.  I will focus on the case of linear periodic equations with a singular drift, which includes (with some regularity assumptions) divergence equations with non-co-normal oscillatory Neumann conditions.  The analysis focuses on an induced integro-differential homogenization problem on the boundary of the domain.  This is joint work with Nestor Guillen.
 
===Jingrui Cheng===
 
Semi-geostrophic system with variable Coriolis parameter.
   
The semi-geostrophic system (abbreviated as SG) is a model of large-scale atmospheric/ocean flows. Previous works about the SG system have been restricted to the case of constant Coriolis force, where we write the equation in "dual coordinates" and solve. This method does not apply for variable Coriolis parameter case. We develop a time-stepping procedure to overcome this difficulty and prove local existence and uniqueness of smooth solutions to SG system. This is joint work with Michael Cullen and Mikhail Feldman.
 
 
===Paul Rabinowitz===
 
On A Double Well Potential System
 
We will discuss an elliptic system of partial differential equations of the form
\[
-\Delta u + V_u(x,u) = 0,\;\;x \in \Omega = \R \times \mathcal{D}\subset \R^n, \;\;\mathcal{D} \; bounded \subset \R^{n-1}
\]
\[
\frac{\partial u}{\partial \nu} = 0 \;\;on \;\;\partial \Omega,
\]
with $u \in \R^m$,\; $\Omega$ a cylindrical domain in $\R^n$, and $\nu$ the outward pointing normal to $\partial \Omega$.
Here $V$ is a double well potential with $V(x, a^{\pm})=0$ and $V(x,u)>0$ otherwise. When $n=1, \Omega =\R^m$ and \eqref{*} is a Hamiltonian system of ordinary differential equations.
When $m=1$, it is a single PDE that arises as an Allen-Cahn model for phase transitions. We will
discuss the existence of solutions of \eqref{*} that are heteroclinic from $a^{-}$ to $a^{+}$ or homoclinic to $a^{-}$,
i.e. solutions that are of phase transition type.
 
This is joint work with Jaeyoung Byeon (KAIST) and Piero Montecchiari (Ancona).
 
===Hong Zhang===
 
On an elliptic equation arising from composite material
 
I will present some recent results on second-order divergence type equations with piecewise constant coefficients. This problem arises in the study of composite materials with closely spaced interface boundaries, and the classical elliptic regularity theory are not applicable. In the 2D case, we show that any weak solution is piecewise smooth without the restriction of the underling domain where the equation is satisfied. This completely answers a question raised by Li and Vogelius (2000) in the 2D case. Joint work with Hongjie Dong.
 
===Aaron Yip===
 
Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media
 
The talk will describe some results on the behavior of solutions of motion by mean curvature in inhomogeneous media. Emphasis will be put on the pinning and de-pinning transition, continuum limit of discrete spin systems and the motion of interface between patterns.
 
 
===Hiroyoshi Mitake===
 
Selection problem for fully nonlinear equations
 
Recently, there was substantial progress on the selection problem on the ergodic problem for Hamilton-Jacobi equations, which was open during almost 30 years. In the talk, I will first show a result on the convex Hamilton-Jacobi equation, then tell important problems which still remain. Next, I will mainly focus on a recent joint work with H. Ishii (Waseda U.), and H. V. Tran (U. Wisconsin-Madison) which is about the selection problem for fully nonlinear, degenerate elliptic partial differential equations. I will present a new variational approach for this problem.
 
===Nestor Guillen===
 
Min-max formulas for integro-differential equations and applications
 
We show under minimal assumptions that a nonlinear operator satisfying what is known as a "global comparison principle" can be represented by a min-max formula in terms of very special linear operators (Levy operators, which involve drift-diffusion and integro-differential terms).  Such type of formulas have been very useful in the theory of second order equations -for instance, by allowing the representation of solutions as value functions for differential games. Applications include results on the structure of Dirichlet-to-Neumann mappings for fully nonlinear second order elliptic equations.
 
===Ryan Denlinger===
 
The propagation of chaos for a rarefied gas of hard spheres in vacuum
 
We are interested in the rigorous mathematical justification of
Boltzmann's equation starting from the deterministic evolution of
many-particle systems. O. E. Lanford was able to derive Boltzmann's
equation for hard spheres, in the Boltzmann-Grad scaling, on a short
time interval. Improvements to the time in Lanford's theorem have so far
either relied on a small data hypothesis, or have been restricted to
linear regimes. We revisit the small data regime, i.e. a sufficiently
dilute gas of hard spheres dispersing into vacuum; this is a regime
where strong bounds are available globally in time. Subject to the
existence of such bounds, we give a rigorous proof for the propagation
of Boltzmann's ``one-sided'' molecular chaos.
 
===Misha Feldman===
 
Shock reflection, free boundary problems and degenerate elliptic equations.
 
Abstract: We will discuss shock reflection problem for compressible gas dynamics, and von Neumann
conjectures on transition between regular and Mach reflections. We will discuss existence of solutions
of regular reflection structure for potential flow equation, and also regularity of solutions, and
properties of the shock curve (free boundary). Our approach is to reduce the shock reflection problem
to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems
will also be discussed, including uniqueness.
The talk is based on the joint works with Gui-Qiang Chen, Myoungjean Bae and Wei Xiang.
 
===Jessica Lin===
 
Optimal Quantitative Error Estimates in Stochastic
Homogenization for Elliptic Equations in Nondivergence Form
 
Abstract: I will present optimal quantitative error estimates in the
stochastic homogenization for uniformly elliptic equations in
nondivergence form. From the point of view of probability theory,
stochastic homogenization is equivalent to identifying a quenched
invariance principle for random walks in a balanced random
environment. Under strong independence assumptions on the environment,
the main argument relies on establishing an exponential version of the
Efron-Stein inequality. As an artifact of the optimal error estimates,
we obtain a regularity theory down to microscopic scale, which implies
estimates on the local integrability of the invariant measure
associated to the process. This talk is based on joint work with Scott
Armstrong.
 
===Sergey Bolotin===
 
Degenerate billiards
 
In an ordinary billiard trajectories of a Hamiltonian system
are elastically reflected  when colliding with a hypersurface (scatterer).
If the scatterer is a submanifold of codimension more than one, then collisions  are rare.
Trajectories with infinite number of collisions form a lower dimensional dynamical system.
Degenerate billiards appear as limits of ordinary billiards and in celestial mechanics.
 
===Moon-Jin Kang===
 
On contraction of large perturbation of shock waves, and inviscid limit problems
 
This talk will start with the relative entropy method to handle the contraction of possibly large perturbations around viscous shock waves of conservation laws. In the case of viscous scalar conservation law in one space dimension, we obtain $L^2$-contraction for any large perturbations of shocks up to a Lipschitz shift depending on time. Such a time-dependent Lipschitz shift should be constructed from dynamics of the perturbation. In the case of multidimensional scalar conservation law, the perturbations of planar shocks are $L^2$-contractive up to a more complicated shift depending on both time and space variable, which solves a parabolic equation with inhomogeneous coefficient and force terms reflecting the perturbation. As a consequence, the $L^2$-contraction property implies the inviscid limit towards inviscid shock waves. At the end, we handle the contraction properties of admissible discontinuities of the hyperbolic system of conservation laws equipped with a strictly convex entropy.

Revision as of 14:26, 15 September 2016

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

Previous PDE/GA seminars

Tentative schedule for Fall 2016

PDE GA Seminar Schedule Fall 2016

date speaker title host(s)
September 12 Daniel Spirn (U of Minnesota) Dipole Trajectories in Bose-Einstein Condensates Kim
September 19 Donghyun Lee (UW-Madison) The Boltzmann equation with specular boundary condition in convex domains Feldman
September 26 Kevin Zumbrun (Indiana) Kim
October 3 Will Feldman (UChicago ) Lin & Tran
October 10 Ryan Hynd (UPenn) Extremal functions for Morrey’s inequality in convex domains Feldman
October 17 Gung-Min Gie (Louisville) Kim
October 24 Tau Shean Lim (UW Madison) TBA Kim & Tran
October 31 Tarek Elgindi ( Princeton) Propagation of Singularities in Incompressible Fluids Lee & Kim
November 7 Adrian Tudorascu (West Virginia) Feldman
November 14 Alexis Vasseur ( UT-Austin) Feldman
November 21 Minh-Binh Tran (UW Madison ) Quantum Kinetic Problems Hung Tran
November 28 ( )
December 5 Brian Weber (University of Pennsylvania) TBA Bing Wang
December 12 David Kaspar (Brown) Tran}

Abstracts
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