# 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.

## Contents

### Previous PDE/GA seminars

### Tentative schedule for Fall 2016

# Seminar Schedule Spring 2016

date | speaker | title | host(s) |
---|---|---|---|

January 25 | Tianling Jin (HKUST and Caltech) | Holder gradient estimates for parabolic homogeneous p-Laplacian equations | Zlatos |

February 1 | Russell Schwab (Michigan State University) | Neumann homogenization via integro-differential methods | Lin |

February 8 | Jingrui Cheng (UW Madison) | Semi-geostrophic system with variable Coriolis parameter | Tran & Kim |

February 15 | Paul Rabinowitz (UW Madison) | On A Double Well Potential System | Tran & Kim |

February 22 | Hong Zhang (Brown) | On an elliptic equation arising from composite material | Kim |

February 29 | Aaron Yip (Purdue university) | Discrete and Continuous Motion by Mean Curvature in Inhomogeneous Media | Tran |

March 7 | Hiroyoshi Mitake (Hiroshima university) | Selection problem for fully nonlinear equations | Tran |

March 15 | Nestor Guillen (UMass Amherst) | Min-max formulas for integro-differential equations and applications | Lin |

March 21 (Spring Break) | |||

March 28 | Ryan Denlinger (Courant Institute) | The propagation of chaos for a rarefied gas of hard spheres in vacuum | Lee |

April 4 | No seminar | ||

April 11 | Misha Feldman (UW) | Shock reflection, free boundary problems and degenerate elliptic equations | |

April 14: 2:25 PM in VV 901-Joint with Probability Seminar | Jessica Lin (UW-Madison) | Optimal Quantitative Error Estimates in Stochastic Homogenization for Elliptic Equations in Nondivergence Form | |

April 18 | Sergey Bolotin (UW) | Degenerate billiards | |

April 21-24, KI-Net conference: Boundary Value Problems and Multiscale Coupling Methods for Kinetic Equations | Link: http://www.ki-net.umd.edu/content/conf?event_id=493 | ||

April 25 | Moon-Jin Kang (UT-Austin) | Kim | |

May 3 (Joint Analysis-PDE seminar ) | Stanley Snelson (University of Chicago) | Seeger & Tran. | |

May 16-20, Conference in Harmonic Analysis in Honor of Michael Christ | Link: https://www.math.wisc.edu/ha_2016/ |

# 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 (UW-Madison)

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.