# Difference between revisions of "PDE Geometric Analysis seminar"

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Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations. | Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations. | ||

They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile. | They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile. | ||

+ | |||

+ | ===Norbert Pozar=== | ||

+ | |||

+ | |||

+ | Viscosity solutions for the crystalline mean curvature flow | ||

+ | |||

+ | In this talk I will present some recent results concerning the analysis | ||

+ | of the level set formulation of the crystalline mean curvature flow. | ||

+ | The crystalline mean curvature, understood as the first variation of an | ||

+ | anisotropic surface energy with an anisotropy whose Wulff shape is a | ||

+ | polytope, is a singular quantity, nonlocal on the flat parts of the | ||

+ | crystal surface. Therefore the level set equation is not a usual PDE and | ||

+ | does not fit into the classical viscosity solution framework. Its | ||

+ | well-posedness in dimensions higher than two was an open problem until | ||

+ | very recently. In a joint work with Yoshikazu Giga (U. of Tokyo), we | ||

+ | introduce a new notion of viscosity solutions for this problem and | ||

+ | establish its well-posedness for compact crystals in an arbitrary | ||

+ | dimension, including a comparison principle and the stability with | ||

+ | respect to approximation by a smooth anisotropic mean curvature flow. |

## Revision as of 16:05, 21 November 2017

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 Spring 2018

## PDE GA Seminar Schedule Fall 2017

## Abstracts

### Mihaela Ifrim

Well-posedness and dispersive decay of small data solutions for the Benjamin-Ono equation

Our goal is to take a first step toward understanding the long time dynamics of solutions for the Benjamin-Ono equation. While this problem is known to be both completely integrable and globally well-posed in $L^2$, much less seems to be known concerning its long time dynamics. We present that for small localized data the solutions have (nearly) dispersive dynamics almost globally in time. An additional objective is to revisit the $L^2$ theory for the Benjamin-Ono equation and provide a simpler, self-contained approach. This is joined work with Daniel Tataru.

### Longjie Zhang

On curvature flow with driving force starting as singular initial curve in the plane

We consider a family of axisymmetric curves evolving by its mean curvature with driving force in the plane. However, the initial curve is oriented singularly at origin. We investigate this problem by level set method and give some criteria to judge whether the interface evolution is fattening or not. In the end, we can classify the solutions into three categories and provide the asymptotic behavior in each category. Our main tools in this paper are level set method and intersection number principle.

### Jaeyoung Byeon

Title: Patterns formation for elliptic systems with large interaction forces

Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.

### Tuoc Phan

Calderon-Zygmund regularity estimates for weak solutions of quasi-linear parabolic equations with an application.

Abstract: In this talk, we first introduce a problem on the existence of global time smooth solutions for a system of cross-diffusion equations. We then recall some classical results on regularity theories, and show that to solve our problem, new results on regularity theory estimates of Calderon-Zygmund type for gradients of solutions to a class of parabolic equations in Lebesgue spaces are required. We then discuss a result on Calderon-Zygmnud type estimate in the concrete setting to solve our mentioned problem regarding the system of cross-diffusion equations. The remaining part of the talk will be focused on some new generalized results on regularity gradient estimates for some general class of quasi-linear parabolic equations. Regularity estimates for gradients of solutions in Lorentz spaces will be presented. Ideas of the proofs for the results are given.

### Hiroyoshi Mitake

Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

### Dongnam Ko

On the emergence of local flocking phenomena in Cucker-Smale ensembles

Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.

### Sameer Iyer

Title: Global-in-x Steady Prandtl Expansion over a Moving Boundary.

Abstract: I will outline the proof that steady, incompressible Navier-Stokes flows posed over the moving boundary, y = 0, can be decomposed into Euler and Prandtl flows globally in the tangential variable, assuming a sufficiently small velocity mismatch. The main obstacles in the analysis center around obtaining sharp decay rates for the linearized profiles and the remainders. The remainders are controlled via a high-order energy method, supplemented with appropriate embedding theorems, which I will present.

### Jingrui Cheng

A 1-D semigeostrophic model with moist convection.

We consider a simplified 1-D model of semigeostrophic system with moisture, which describes moist convection in a single column in the atmosphere. In general, the solution is non-continuous and it is nontrivial part of the problem to find a suitable definition of weak solutions. We propose a plausible definition of such weak solutions which describes the evolution of the probability distribution of the physical quantities, so that the equations hold in the sense of almost everywhere. Such solutions are constructed from a discrete scheme which obeys the physical principles. This is joint work with Mike Cullen, together with Bin Cheng, John Norbury and Matthew Turner.

### Donghyun Lee

We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. Moreover we prove an exponential convergence of distribution function toward the global Maxwellian.

### Myoungjean Bae

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler-Poisson system.

I will present a recent result on the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=angular momentum density) component. This talk is based on a joint work with S. Weng (Wuhan University, China).

### Jingchen Hu

Shock Reflection and Diffraction Problem with Potential Flow Equation

In this talk, we will present our work on nonsymmetric shock reflection and diffraction problem, the equation concerned is potential flow equation, which is a simplification of Euler System, mainly based on the assumption that flow has no vortex. We showed in both nonsymmetric reflection case and diffraction case, that physically admissible solution does not exist. This implies that the formation of vortex is essential to maintain the structural stability of shock reflection and diffraction.

### Xiaoqin Guo

Quantitative homogenization and Harnack inequality for a degenerate discrete nondivergence form random operator.

In the d-dimensional integer lattice $\mathbb Z^d$, $d\ge 2$, we consider a discrete non-divergence form difference operator $$ L_a u(x)=\sum_{i=1}^d a_i(x)[u(x+e_i)+u(x-e_i)-2u(x)] $$ where $a(x)=diag(a_1(x),..., a_d(x)), x\in\mathbb Z^d$ are random nonnegative diagonal matrices which are identically distributed and independent and with a positive expectation. A difficulty in studying this problem is that coefficients are allowed to be zero. In this talk, using random walks in random media and its percolative structure, we will present a Harnack inequality and quantitative homogenization result for this random operator. Joint work with N.Berger, M.Cohen and J.-D. Deuschel.

### Ru-Yu Lai

Inverse problems for Maxwell's equations and its application.

This talk will illustrate the application of complex geometrical optics (CGO) solutions to Maxwell's equations. First, I will explain the increasing stability behavior of coefficients for Maxwell equations. In particular, by using CGO solutions, the stability estimate of the conductivity is improving when frequency is growing. Second, I will describe the construction of new families of accelerating and almost nondiffracting beams for Maxwell's equations. They have the form of wave packets that propagate along circular trajectories while almost preserving a trasverse intensity profile.

### Norbert Pozar

Viscosity solutions for the crystalline mean curvature flow

In this talk I will present some recent results concerning the analysis of the level set formulation of the crystalline mean curvature flow. The crystalline mean curvature, understood as the first variation of an anisotropic surface energy with an anisotropy whose Wulff shape is a polytope, is a singular quantity, nonlocal on the flat parts of the crystal surface. Therefore the level set equation is not a usual PDE and does not fit into the classical viscosity solution framework. Its well-posedness in dimensions higher than two was an open problem until very recently. In a joint work with Yoshikazu Giga (U. of Tokyo), we introduce a new notion of viscosity solutions for this problem and establish its well-posedness for compact crystals in an arbitrary dimension, including a comparison principle and the stability with respect to approximation by a smooth anisotropic mean curvature flow.