# Difference between revisions of "PDE Geometric Analysis seminar"

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| Theodora Bourni (UT Knoxville) | | Theodora Bourni (UT Knoxville) | ||

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Abstract: I will present the results of my recent work with Gui-Qiang Chen on the Euler equations in the conditions of special relativity. I will show how the theory of compensated compactness may be used to obtain the existence of entropy solutions to this system. Moreover, it is expected that as the light speed grows to infinity, solutions to the relativistic Euler equations will converge to their classical (Newtonian) counterparts. The theory we develop is also sufficient to demonstrate this convergence rigorously. | Abstract: I will present the results of my recent work with Gui-Qiang Chen on the Euler equations in the conditions of special relativity. I will show how the theory of compensated compactness may be used to obtain the existence of entropy solutions to this system. Moreover, it is expected that as the light speed grows to infinity, solutions to the relativistic Euler equations will converge to their classical (Newtonian) counterparts. The theory we develop is also sufficient to demonstrate this convergence rigorously. | ||

+ | |||

+ | ===Theodora Bourni=== | ||

+ | Title: Polygonal Pancakes | ||

+ | |||

+ | Abstract: We study ancient collapsed solutions to mean curvature flow, $\{M^n_t\}_{t\in(-\infty,0)}$, in terms of their squash down: $\Omega_*=\lim_{t\to-\infty}\frac{1}{-t} M_t$. We show that $\Omega_*$ must be a convex body which circumscribes $S^1$ and for every such $\Omega_*$ we construct a solution with this prescribed squash down. Our analysis includes non-compact examples, in which setting we disprove a conjecture of White stating that all eternal solutions must be translators. This is joint work with Langford and Tinaglia. | ||

===Zhiyan Ding=== | ===Zhiyan Ding=== |

## Revision as of 10:12, 26 February 2020

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 2020-Spring 2021

## PDE GA Seminar Schedule Fall 2019-Spring 2020

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

Sep 9 | Scott Smith (UW Madison) | Recent progress on singular, quasi-linear stochastic PDE | Kim and Tran |

Sep 14-15 | AMS Fall Central Sectional Meeting https://www.ams.org/meetings/sectional/2267_program.html | ||

Sep 23 | Son Tu (UW Madison) | State-Constraint static Hamilton-Jacobi equations in nested domains | Kim and Tran |

Sep 28-29, VV901 | https://www.ki-net.umd.edu/content/conf?event_id=993 | Recent progress in analytical aspects of kinetic equations and related fluid models | |

Oct 7 | Jin Woo Jang (Postech) | On a Cauchy problem for the Landau-Boltzmann equation | Kim |

Oct 14 | Stefania Patrizi (UT Austin) | Dislocations dynamics: from microscopic models to macroscopic crystal plasticity | Tran |

Oct 21 | Claude Bardos (Université Paris Denis Diderot, France) | From d'Alembert paradox to 1984 Kato criteria via 1941 1/3 Kolmogorov law and 1949 Onsager conjecture | Li |

Oct 25-27, VV901 | https://www.ki-net.umd.edu/content/conf?event_id=1015 | Forward and Inverse Problems in Kinetic Theory | Li |

Oct 28 | Albert Ai (UW Madison) | Two dimensional gravity waves at low regularity: Energy estimates | Ifrim |

Nov 4 | Yunbai Cao (UW Madison) | Vlasov-Poisson-Boltzmann system in Bounded Domains | Kim and Tran |

Nov 18 | Ilyas Khan (UW Madison) | The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension | Kim and Tran |

Nov 25 | Mathew Langford (UT Knoxville) | Concavity of the arrival time | Angenent |

Dec 9 - Colloquium (4-5PM) | Hui Yu (Columbia) | TBA | Tran |

Feb. 3 | Philippe LeFloch (Sorbonne University and CNRS) | Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions | Feldman |

Feb. 10 | Joonhyun La (Stanford) | On a kinetic model of polymeric fluids | Kim |

Feb 17 | Yannick Sire (JHU) | Minimizers for the thin one-phase free boundary problem | Tran |

Feb 19 - Colloquium (4-5PM) | Zhenfu Wang (University of Pennsylvania) | Quantitative Methods for the Mean Field Limit Problem | Tran |

Feb 24 | Matthew Schrecker (UW Madison) | Existence theory and Newtonian limit for 1D relativistic Euler equations | Feldman |

March 2 | Theodora Bourni (UT Knoxville) | Polygonal Pancakes | Angenent |

March 3 -- Analysis seminar | William Green (Rose-Hulman Institute of Technology) | Dispersive estimates for the Dirac equation | Betsy Stovall |

March 9 | Ian Tice (CMU) | TBA | Kim |

March 16 | No seminar (spring break) | TBA | Host |

March 23 | Jared Speck (Vanderbilt) | TBA | Schrecker |

March 30 | Huy Nguyen (Brown) | TBA | Kim and Tran |

April 6 | Zhiyan Ding (UW Madison) | Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis | Kim and Tran |

April 13 | Hyunju Kwon (IAS) | TBA | Kim |

April 20 | Adrian Tudorascu (WVU) | On the Lagrangian description of the Sticky Particle flow | Feldman |

April 27 | Christof Sparber (UIC) | TBA | Host |

May 18-21 | Madison Workshop in PDE 2020 | TBA | Tran |

## Abstracts

### Scott Smith

Title: Recent progress on singular, quasi-linear stochastic PDE

Abstract: This talk with focus on quasi-linear parabolic equations with an irregular forcing . These equations are ill-posed in the traditional sense of distribution theory. They require flexibility in the notion of solution as well as new a priori bounds. Drawing on the philosophy of rough paths and regularity structures, we develop the analytic part of a small data solution theory. This is joint work with Felix Otto, Hendrik Weber, and Jonas Sauer.

### Son Tu

Title: State-Constraint static Hamilton-Jacobi equations in nested domains

Abstract: We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}$ in $\mathbb R^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k \in \mathbb N$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega=\bigcup_k \Omega_k$. In many cases, the rates obtained are proven to be optimal (it's a joint work with Yeoneung Kim and Hung V. Tran).

### Jin Woo Jang

Title: On a Cauchy problem for the Landau-Boltzmann equation

Abstract: In this talk, I will introduce a recent development in the global well-posedness of the Landau equation (1936) in a general smooth bounded domain, which has been a long-outstanding open problem. This work proves the global stability of the Landau equation in an $L^\infty_{x,v}$ framework with the Coulombic potential in a general smooth bounded domain with the specular reflection boundary condition for initial perturbations of the Maxwellian equilibrium states. Our methods consist of the generalization of the well-posedness theory for the kinetic Fokker-Planck equation (HJV-2014, HJJ-2018) and the extension of the boundary value problem to a whole space problem, as well as the use of a recent extension of De Giorgi-Nash-Moser theory for the kinetic Fokker-Planck equations (GIMV-2016) and the Morrey estimates (BCM-1996) to further control the velocity derivatives, which ensures the uniqueness. This is a joint work with Y. Guo, H. J. Hwang, and Z. Ouyang.

### Stefania Patrizi

Title: Dislocations dynamics: from microscopic models to macroscopic crystal plasticity

Abstract: Dislocation theory aims at explaining the plastic behavior of materials by the motion of line defects in crystals. Peierls-Nabarro models consist in approximating the geometric motion of these defects by nonlocal reaction-diffusion equations. We study the asymptotic limit of solutions of Peierls-Nabarro equations. Different scalings lead to different models at microscopic, mesoscopic and macroscopic scale. This is joint work with E. Valdinoci.

### Claude Bardos

Title: From the d'Alembert paradox to the 1984 Kato criteria via the 1941 $1/3$ Kolmogorov law and the 1949 Onsager conjecture

Abstract: Several of my recent contributions, with Marie Farge, Edriss Titi, Emile Wiedemann, Piotr and Agneska Gwiadza, were motivated by the following issues: The role of boundary effect in mathematical theory of fluids mechanic and the similarity, in presence of these effects, of the weak convergence in the zero viscosity limit and the statistical theory of turbulence. As a consequence, I will recall the Onsager conjecture and compare it to the issue of anomalous energy dissipation.

Then I will give a proof of the local conservation of energy under convenient hypothesis in a domain with boundary and give supplementary condition that imply the global conservation of energy in a domain with boundary and the absence of anomalous energy dissipation in the zero viscosity limit of solutions of the Navier-Stokes equation in the presence of no slip boundary condition.

Eventually the above results are compared with several forms of a basic theorem of Kato in the presence of a Lipschitz solution of the Euler equations and one may insist on the fact that in such case the the absence of anomalous energy dissipation is {\bf equivalent} to the persistence of regularity in the zero viscosity limit. Eventually this remark contributes to the resolution of the d'Alembert Paradox.

### Albert Ai

Title: Two dimensional gravity waves at low regularity: Energy estimates

Abstract: In this talk, we will consider the gravity water wave equations in two space dimensions. Our focus is on sharp cubic energy estimates and low regularity solutions. Precisely, we will introduce techniques to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier cubic estimates of Hunter-Ifrim-Tataru, while preserving their scale invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness. This is joint work with Mihaela Ifrim and Daniel Tataru.

### Ilyas Khan

Title: The Uniqueness of Asymptotically Conical Self-Shrinkers in High Codimension.

Abstract: In this talk, we will consider self-shrinking solitons of the mean curvature flow that are smoothly asymptotic to a Riemannian cone in $\mathbb{R}^n$. In 2011, L. Wang proved the uniqueness of self-shrinking ends asymptotic to a cone $C$ in the case of hypersurfaces (codimension 1) by using a backwards uniqueness result for the heat equation due to Escauriaza, Sverak, and Seregin. Later, J. Bernstein proved the same fact using purely elliptic methods. We consider the case of self-shrinkers in high codimension, and outline how to prove the same uniqueness result in this significantly more general case, by using geometric arguments and extending Bernstein’s result.

### Mathew Langford

Title: Concavity of the arrival time

Abstract: We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a novel concavity maximum principle argument to the corresponding level set flow equations. In particular, this yields a short proof of Hamilton’s differential Harnack inequality for mean curvature flow and, more generally, Andrews’ differential Harnack inequalities for certain “$\alpha$-inverse-concave” flows.

### Philippe LeFloch

Title: Nonlinear stability of self-gravitating matter under low decay and weak regularity conditions

Abstract: I will present recent progress on the global evolution problem for self-gravitating matter. (1) For Einstein's constraint equations, motivated by a scheme proposed by Carlotto and Schoen I will show the existence of asymptotically Euclidean Einstein spaces with low decay; joint work with T. Nguyen.

(2) For Einstein's evolution equations in the regime near Minkowski spacetime, I will show the global nonlinear stability of massive matter fields; joint work with Y. Ma.

(3) For the colliding gravitational wave problem, I will show the existence of weakly regular spacetimes containing geometric singularities across which junction conditions are imposed; joint work with B. Le Floch and G. Veneziano.

### Joonhyun La

Title: On a kinetic model of polymeric fluids

Abstract: In this talk, we prove global well-posedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multi-scale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.

### Yannick Sire

Title: Minimizers for the thin one-phase free boundary problem

Abstract: We consider the thin one-phase free boundary problem, associated to minimizing a weighted Dirichlet energy of thefunction in the half-space plus the area of the positivity set of that function restricted to the boundary. I will provide a rather complete picture of the (partial ) regularity of the free boundary, providing content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight related to an anomalous diffusion on the boundary. The approach does not follow the standard one introduced in the seminal work of Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE. This opens several directions of research that I will try to describe.

### Matthew Schrecker

Title: Existence theory and Newtonian limit for 1D relativistic Euler equations

Abstract: I will present the results of my recent work with Gui-Qiang Chen on the Euler equations in the conditions of special relativity. I will show how the theory of compensated compactness may be used to obtain the existence of entropy solutions to this system. Moreover, it is expected that as the light speed grows to infinity, solutions to the relativistic Euler equations will converge to their classical (Newtonian) counterparts. The theory we develop is also sufficient to demonstrate this convergence rigorously.

### Theodora Bourni

Title: Polygonal Pancakes

Abstract: We study ancient collapsed solutions to mean curvature flow, $\{M^n_t\}_{t\in(-\infty,0)}$, in terms of their squash down: $\Omega_*=\lim_{t\to-\infty}\frac{1}{-t} M_t$. We show that $\Omega_*$ must be a convex body which circumscribes $S^1$ and for every such $\Omega_*$ we construct a solution with this prescribed squash down. Our analysis includes non-compact examples, in which setting we disprove a conjecture of White stating that all eternal solutions must be translators. This is joint work with Langford and Tinaglia.

### Zhiyan Ding

Title: Ensemble Kalman Sampling: well-posedness, mean-field limit and convergence analysis

Abstract: Ensemble Kalman Sampling (EKS) is a method to find iid samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. In this talk, I will focus on the continuous version of EKS with linear forward map, a coupled SDE system. I will talk about its well-posedness and justify its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution.

### Adrian Tudorascu

Title: On the Lagrangian description of the Sticky Particle flow

Abstract: R. Hynd has recently proved that for absolutely continuous initial velocities the Sticky Particle system admits solutions described by monotone flow maps in Lagrangian coordinates. We present a generalization of this result to general initial velocities and discuss some consequences. (This is based on ongoing work with M. Suder.)