# Difference between revisions of "PDE Geometric Analysis seminar"

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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. | 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. |

## Revision as of 11:11, 28 March 2016

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 | |||

April 11 | Misha Feldman (UW) | ||

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

April 18 | Sergey Bolotin (UW) | Degenerate billiards in celestial mechanics | |

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 2 |

# 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.*