# 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

# 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## Tianling JinHolder 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 SchwabNeumann 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 ChengSemi-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 RabinowitzOn 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 ZhangOn 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 YipDiscrete 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 MitakeSelection 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 GuillenMin-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 DenlingerThe 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 ## Misha FeldmanShock 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 LinOptimal 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 BolotinDegenerate 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 KangOn 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. |