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 Fall 2018-Spring 2019

date	speaker	title	host(s)
August 31 (FRIDAY),	Julian Lopez-Gomez (Complutense University of Madrid)	The theorem of characterization of the Strong Maximum Principle	Rabinowitz
September 10,	Hiroyoshi Mitake (University of Tokyo)	On approximation of time-fractional fully nonlinear equations	Tran
September 12 and September 14,	Gunther Uhlmann (UWash)	TBA	Li
September 17,	Changyou Wang (Purdue)	Some recent results on mathematical analysis of Ericksen-Leslie System	Tran
Sep 28, Colloquium	Gautam Iyer (CMU)	Stirring and Mixing	Thiffeault
October 1,	Matthew Schrecker (UW)	Finite energy methods for the 1D isentropic Euler equations	Kim and Tran
October 8,	Anna Mazzucato (PSU)	On the vanishing viscosity limit in incompressible flows	Li and Kim
October 15,	Lei Wu (Lehigh)	Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects	Kim
October 22,	Annalaura Stingo (UCD)	Global existence of small solutions to a model wave-Klein-Gordon system in 2D	Mihaela Ifrim
October 29,	Yeon-Eung Kim (UW)	Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties	Kim and Tran
November 5,	Albert Ai (UC Berkeley)	Low Regularity Solutions for Gravity Water Waves	Mihaela Ifrim
Nov 7 (Wednesday), Colloquium	Luca Spolaor (MIT)	(Log)-Epiperimetric Inequality and the Regularity of Variational Problems	Feldman
December 3,	Trevor Leslie (UW)	TBA	Kim and Tran
December 10, Time: 2:25, Room: B223 Van Vleck	Serena Frederico (MIT)	TBA	Mihaela Ifrim
January 28,	( )	TBA
Time: TBD in February,	Xiaoqin Guo (UW)	TBA	Kim and Tran
Time: TBD,	Jessica Lin (McGill University)	TBA	Tran
March 4	Vladimir Sverak (Minnesota)	TBA(Wasow lecture)	Kim
March 11	Jonathan Luk (Stanford)	TBA	Kim
March 18,	Spring recess (Mar 16-24, 2019)
April 15,	Yao Yao (Gatech)	TBA	Tran
April 29,	( )	TBA

Abstracts

Julian Lopez-Gomez

Title: The theorem of characterization of the Strong Maximum Principle

Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.

Hiroyoshi Mitake

Title: On approximation of time-fractional fully nonlinear equations

Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)

Changyou Wang

Title: Some recent results on mathematical analysis of Ericksen-Leslie System

Abstract: The Ericksen-Leslie system is the governing equation that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.

Matthew Schrecker

Title: Finite energy methods for the 1D isentropic Euler equations

Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.

Anna Mazzucato

Title: On the vanishing viscosity limit in incompressible flows

Abstract: I will discuss recent results on the analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions. I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.

Lei Wu

Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects

Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.

Annalaura Stingo

Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D

Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms » . Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.

Yeon-Eung Kim

Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties

A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.

Albert Ai

Title: Low Regularity Solutions for Gravity Water Waves

Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.