# PDE Seminar via Zoom

Updated: 13 hours ago

**Organizer:** Quoc-Hung Nguyen, Institute of Mathematical Sciences, ShanghaiTech University

### The Seminar's purpose is to promote and influence more cooperation, understanding, and collaboration among scientists working in Partial Differential Equations.

Due to the coronavirus pandemic and in a time of social distancing, PDE Seminar via Zoom aims to provide such a venue for the PDE community, and keep as many researchers as possible stay connected.

**How to use Zoom:**

The seminar will take place over Zoom. Please download and setup Zoom via link: https://zoom.us/download

Please log in with your real name.

About one day before the seminar, the link, meeting ID and password to join the Zoom meeting will be attached here.

Audiences can use "

**Raise Hand"**on the**chat**window

**Program:** (You can see titles and abstracts in __this pdf file__ __)__

__1. April 9th, 2020__

**9 pm Beijing Time (GMT+8):**

**Speaker: **Elia Brué, SNS, PISA.

**Title: ***Classical non-uniqueness of characteristic curves associated to Sobolev vector fields*

__Video link__ and __Slides__

*10 pm Beijing Time (GMT+8):*

**Speaker:** Hyunju Kwon, IAS, Princeton

**Title:** *Strong ill-posedness of the logarithmically regularized 2D Euler equations in the borderline Sobolev space*

**11 pm Beijing Time (GMT+8):**

**Speaker: **Zhenfu Wang, University of Pennsylvania

**Title:** *Quantitative Methods for the Mean Field Limit Problem*

__Video link__ and __Slides__

__2. April 16th, 2020__

**8:45 pm***Beijing Time (GMT+8):*

** **Welcome Addresses by Xiuxiong Chen, Founding Director of IMS

__Video link__ and __Slides__

**9:15 pm Beijing Time (GMT+8):**

**Speaker:** Weiren Zhao, NYU Abu Dhabi

**Title:*** **Inviscid damping for a class of monotone shear flow*

__Video link__ and __Slides__

**10:05 pm Beijing Time (GMT+8):**

**Speaker:** Xavier Ros-Oton, Universität Zürich
**Title: ***Generic regularity of free boundaries for the obstacle problem*

__Video link__ and __Slides__

**11 pm Beijing Time (GMT+8):**

**Speaker:** Hui Yu, Columbia University
**Title:** *Regularity of the singular set in the fully nonlinear obstacle problem*

__Video link__ and __Slides__

__3. April 23rd, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker: ** Elio Marconi__,__ University of Basel

**Title:*** Regularity estimates for the flow of BV autonomous divergence-free planar vector fields*

__Video link__ and __Slides__

*10 pm Beijing Time (GMT+8):*

**Speaker: ** Charles Collot__,__ Courant institute of mathematical Sciences
**Title:*** On the derivation of the homogeneous kinetic wave equation*
__Video link__ and __Slides__

*11 pm Beijing Time (GMT+8):*

**Speaker:** Jacob Bedrossian__,__ University of Maryland

**Title:** *The power spectrum of passive scalar turbulence in the Batchelor regime*

__Video link__ and __Slides__

__4. April 30th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker: **Tobias Barker*,* École normale supérieure
**Title:** *Quantitative estimates for the Navier-Stokes equations via spatial concentration*

*10 pm Beijing Time (GMT+8):*

**Speaker: **Camillo De Lellis, IAS, Princeton

**Title: ***The oriented Plateau problem and a question of Almgren*

__Video Link__ and __Slides__

*11 pm Beijing Time (GMT+8):*

**Speaker: **Yu Deng, University of Southern California

**Title:** Derivation of the wave kinetic equation

__Video Link__ and __Slides__

__5. May 7th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker: **Pierre Raphael, University of Cambridge

**Title:** On blow up for the defocusing NLS and three dimensional viscous compressible fluids

*10 pm Beijing Time (GMT+8):*

**Speaker: ***Guido De Philippis*, Courant institute of mathematical Sciences

**Title:** *Regularity of the free boundary for the two-phase Bernoulli problem*

*11 pm Beijing Time (GMT+8):*

**Speaker: **Didier Bresch, Université Savoie Mont-Blanc

**Title:** *On the stationary compressible Navier-Stokes equations*

__Video Link__ and __Slides__

__6. May 14th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker:** In-Jee Jeong, Korea Institute for Advanced Study (KIAS)

**Title:** Well-posedness for the axisymmetric Euler equations

*10 pm Beijing Time (GMT+8):*

**Speaker: **Alexandru Ionescu, Princeton University

**Title:** Nonlinear stability of vortices and shear flows

__Slides__ and __Video link__

*11 pm Beijing Time (GMT+8):*

**Speaker:**Theodore Drivas, Princeton University

**Title:** Quasisymmetric plasma equilibria with small forcing

__7. May 21st, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker:** Isabelle Gallagher__,__ École normale supérieure

**Title:** From Newton to Boltzmann, fluctuations and large deviations

*10 pm Beijing Time (GMT+8):*

**Speaker:** Maria Colombo__,__ *Ecole polytechnique fédérale de Lausanne (EPFL)*

**Title:** Weak solutions of the Navier-Stokes equations may be smooth for a.e. time

*11 pm Beijing Time (GMT+8):*

**Speaker:** Julian Fischer__,__ Institute of Science and Technology Austria (IST Austria)

**Title:** Weak-strong uniqueness principles for interface evolution problems in fluid mechanics and geometry

__8. May 28th, 2020__

__Zoom Link__ and ID: 842 0656 6109, password: 050224

*9 pm Beijing Time (GMT+8):*

**Speaker: **Toan T. Nguyen__,__ Penn State University

**Title:** Landau damping and Plasma echoes

**Abstract: **The talk presents an elementary proof of the nonlinear Landau damping for analytic and Gevrey data that was first obtained by Mouhot and Villani and subsequently extended by Bedrossian, Masmoudi, and Mouhot. The construction of an infinite cascade of plasma echoes, that do not belong to the analytic or Gevrey classes, but do, nonetheless, exhibit damping phenomena for large times, will also be presented. This is a joint work with Emmanuel Grenier (ENS Lyon) and Igor Rodnianski (Princeton).

*10 pm Beijing Time (GMT+8):*

**Speaker:** Pierre-Emmanuel Jabin, University of Maryland

**Title:** Large stochastic systems of interacting particles

**Abstract:** I will present some recent results, obtained with D. Bresch and Z. Wang, on large stochastic many-particle or multi-agent systems. Because such systems are conceptually simple but exhibit a wide range of emerging macroscopic behaviors, they are now employed in a large variety of applications from Physics (plasmas, galaxy formation...) to the Biosciences, Economy, Social Sciences...

The number of agents or particles is typically quite large, with 10^{20}-10^{25} particles in many Physics settings for example and just as many equations. Analytical or numerical studies of such systems are potentially very complex leading to the key question as to whether it is possible to reduce this complexity, notably thanks to the notion of propagation of chaos (agents remaining almost uncorrelated).

To derive this propagation of chaos, we have introduced a novel analytical method, which led to the resolution of two long-standing conjectures:

The quantitative derivation of the 2-dimensional incompressible Navier-Stokes system from the point vortices dynamics;

The derivation of the mean-field limit for attractive singular interactions such as in the Keller-Segel model for chemotaxis and some Coulomb gases.

*11 pm Beijing Time (GMT+8):*

* ***Speaker: **Thomas Yizhao Hou**, **California Institute of Technology

**Title:** Recent Progress on Singularity Formation of 3D Euler Equations and Related Models

**Abstract: **Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. We first review the numerical evidence of finite time singularity for 3D axisymmetric Euler equations by Luo and Hou. The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder. We then present a novel method of analysis and prove that the 1D HL model develops finite time self-similar singularity. We also apply this method of analysis to prove finite time self-similar blowup of the original De Gregorio model for some smooth initial data on the real line with compact support. Self-similar blowup results for the generalized De Gregorio model for the entire range of parameter on the real line or on a circle have been obtained for Holder continuous initial data with compact support. Finally, we report our recent progress in analyzing the finite time singularity of the axisymmetric 3D Euler equations with initial data considered by Luo and Hou.

__9. June 4th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker:** Michele Coti Zelati, Imperial College London

**Title:** Inviscid damping and enhanced dissipation in 2d fluids

**Abstract:** We review some recent results on the asymptotic stability of stationary solutions to the two-dimensional Euler and Navier-Stokes equations of incompressible flows. In many cases, sharp decay rates for the linearized problem imply some sort of nonlinear asymptotic stability, both in the Euler equations (through the so-called inviscid damping) and the Navier-Stokes equations (undergoing enhanced dissipation). However, we will see that in the case of the 2D square periodic domain, the so-called Kolmogorov flow exhibits much more complex behavior: in particular, linear asymptotic stability holds, while nonlinear asymptotic stability is not true even for analytic perturbations.

*10 pm Beijing Time (GMT+8):*

**Speaker:** Thomas Alazard, École Normale Supérieure de Paris-Saclay

**Title:** Entropies of free surface flows in fluid dynamics

**Abstract:** I will discuss recent works with Didier Bresch, Nicolas Meunier and Didier Smets about the dynamics of a free surface transported by an incompressible flow obeying Darcy’s law. I will consider the Hele-Shaw and Mullins-Sekerka equations, as well as the thin-film and Boussinesq equations. For these equations, I will present monotonicity properties of different natures : maximum principles, Lyapunov functionals and entropies. The analysis is based on exact identities which in turn allow to study the Cauchy problem for classical solutions in any subcritical Sobolev spaces.

*11 pm Beijing Time (GMT+8):*

**Speaker:** Phan Thanh Nam, LMU Munich

**Title: **Derivation of the Bose-Einstein condensation for trapped bosons

**Abstract: **TBA

__10. June 11th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker:** Zhifei Zhang, Peking university

**Title:** Transition threshold for the 3D Couette flow in a finite channel

**Abstract:** The plane Couette flow is linearly stable for any Reynolds number. However, it could become nonlinearly unstable and transition to turbulence for small but finite perturbations at high Reynolds number. This is so-called Sommerfeld paradox.
One resolution of this paradox is to study the transition threshold problem, which is concerned with how much disturbance will lead to the instability of the flow and the dependence of disturbance on the Reynolds number. In a joint work with Qi Chen and Dongyi Wei, we showed that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0{Re}^{-1}$ for some $c_0>0$ independent of $Re$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow in the $L^\infty$ sense, and rapidly converges to a streak solution for $t\gtrsim Re^{1/3}$ due to the mixing-enhanced dissipation effect. This result confirms the transition threshold conjecture proposed by Trefethen et al.(Science, 261(1993), 578-584) for the 3D Couette flow in a finite channel with non-slip boundary condition.

*10 pm Beijing Time (GMT+8):*

**Speaker:** Clément Mouhot, University of Cambridge

**Title:** Unified approach to fluid approximation of linear kinetic equations with heavy tails

**Abstract:** The rigorous fluid approximation of linear kinetic equations was first obtained in the late 70s when the equilibrium distribution decays faster than polynomials. In this case the limit is a diffusion equation. In the case of heavy tail equilibrium distribution (with infinite variance), the first rigorous derivation was obtained in 2011 in my joint paper with Mellet and Mischler, in the case of scattering operators. The limit shows then anomalous diffusion; it is governed by a fractional diffusion equation. Lebeau and Puel proved last year the first similar result for Fokker-Planck operator, in dimension 1 and assuming that the equilibrium distribution has finite mass. Fournier and Tardif gave an alternative probabilistic proof, more general (covering any dimension and infinite-mass equilibrium distribution) but non constructive. We present a unified quantitative PDE approach that obtains constructively the limit for Fokker-Planck operators in dimensions greater than 2, but also recovers and unifies the previous works. This is a joint work with Emeric Bouin (Université Paris-Dauphine).

*11 pm Beijing Time (GMT+8):*

**Speaker:** Luis Silvestre, University of Chicago

**Title:** Regularity estimates for the Boltzmann equation without cutoff.

**Abstract:** We study the regularization effect of the inhomogeneous Boltzmann equation without cutoff. We obtain a priori estimates for all derivatives of the solution depending only on bounds of the hydrodynamic quantities: mass density, energy density and entropy density. As a consequence, a classical solution to the equation may fail to exists after certain time T only if at least one of these hydrodynamic quantities blows up. Our analysis applies to the case of moderately soft and hard potentials. We use methods that originated in the study of nonlocal elliptic equations: a weak Harnack inequality in the style of De Giorgi, and a Schauder-type estimate.

__11. June 18th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker:** Fabio Pusateri, University of Toronto ( To be confirmed)

**Title:** TBA

**Abstract:** TBA

*10 pm Beijing Time (GMT+8):*

**Speaker:** Max Engelstein, University of Minnesota

**Title:** An Epiperimetric Approach to Isolated Singularities

**Abstract:** The presence of singular points (i.e. points around which the object in question does not look flat at any scale) is inevitable in most minimization problems. One fundamental question is whether minimizers have a unique tangent object at singular points i.e., is the minimizer increasingly well approximated by some other minimizing object as we “zoom in” at a singular point. This question has been investigated with varying degrees of success in the settings of minimal surfaces, harmonic maps and obstacle problems amongst others.

In this talk, we will present an uniqueness of blowups result for minimizers of the Alt-Caffarelli functional. In particular, we prove that the tangent object is unique at isolated singular points in the free boundary. Our main tool is a new approach to proving (log-)epiperimetric inequalities at isolated singularities. This epiperimetric inequality differs from previous ones in that it holds without any additional assumptions on the symmetries of the tangent object.

If we have time, we will also discuss how this method allows us to recover some uniqueness of blow-ups results in the minimal surfaces setting, particularly those of Allard-Almgren (’81) and Leon Simon (’83). This is joint work with Luca Spolaor (UCSD) and Bozhidar Velichkov (U. Napoli).

*11 pm Beijing Time (GMT+8):*

**Speaker:** Benoit Pausader, Brown University

**Title:** TBA

**Abstract:** TBA

__12. June 25th, 2020__

*9 pm Beijing Time (GMT+8):*

** Speaker: **Pierre Gilles Lemarié-Rieusset, Laboratoire de Mathématiques et Modélisation d'Évry

**Title:** TBA

**Abstract:** TBA

*10 pm Beijing Time (GMT+8):*

** Speaker: **Robert M. Strain, University of Pennsylvania

**Title:** TBA

**Abstract:** TBA

*11 pm Beijing Time (GMT+8):*

**Speaker:** Sung-Jin Oh, University of California Berkeley

**Title:** On the Cauchy problem for the Hall magnetohydrodynamics

**Abstract:** In this talk, I will describe a recent series of work with I.-J. Jeong on the Hall MHD equation without resistivity. This PDE, first investigated by the applied mathematician M. J. Lighthill, is a one-fluid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic field) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic field equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bicharacteristic flow associated with the appropriate principal symbol.

__13. July 2nd, 2020__

*9 pm Beijing Time (GMT+8)*

**Speaker: **Massimiliano Berti, SISSA

**Title:** TBA

**Abstract:**TBA

*10 pm Beijing Time (GMT+8)*

**Speaker:** Nicolas Burq, Institut de Mathématiques d'Orsay

**Title:** Control for wave equations: revisiting the geometric control condition 30 years later

**Abstract:** Following the pioneering work by Bardos-Lebeau and Rauch, the property of controllability for the wave equation has been intensively studied, mainly in a smooth framework (smooth metric and smooth domain). In this lecture, and I shall present some new results on observability/control for the wave equation with rough coefficients. This question leads to some interesting ODE questions for vector fields with only continuous coefficients.

This is joint work B. Dehman (Université Tunis El Manar) and J. Le Rousseau (Université Paris 13).

*11 pm Beijing Time (GMT+8)*

**Speaker:** Huy Nguyen, Brown University

**Title:** TBA

**Abstract:** TBA

__14. July 9th, 2020__

*9 pm Beijing Time (GMT+8):*

**Speaker: **Xavier Tolsa, Autonomous University of Barcelona

**Title:** Unique continuation at the boundary for harmonic functions

**Abstract: **In a work from 1991 Fang-Hua Lin asked the following question. Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Let $u$ be a

function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes et $\Sigma \subset \partial\Omega$ and moreover assume that

the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure. Is it true that then $u$ is identically zero?

Up to now, the answer was known to be positive for $C^1$-Dini domains, by results of Adolfsson-Escauriaza (1997) and Kukavica-Nystrom (1998). In this talk I will explain a recent work where I show that the result also holds for Lipschitz domains with small Lipschitz constant, and thus in particular for general $C^1$ domains.

*10 pm Beijing Time (GMT+8)*

**Speaker: ** Bogdan-Vasile Matioc, University of Regensburg

**Title:** The Muskat problem in subcritical Lp-Sobolev spaces

**Abstract: **The Muskat problem is a classical mathematical model which describes the motion of two immiscible and incompressible Newtonian fluids in an homogeneous porous medium. The mathematical model posed in the entire plane can be formulated as an evolution equation for the function that parametrizes the free boundary between the fluids. When neglecting surface tension effects, the evolution equation is fully nonlinear and nonlocal and it involves singular integral operators defined by kernels that depend nonlinearly on the unknown. We prove that the evolution problem is of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. Based upon this feature we establish the well posedness of the Muskat problem in all subcritical Lp-Sobolev spaces together with a parabolic smoothing property. This is a joint work with Helmut Abels.

*11 pm Beijing Time (GMT+8)*

**Speaker:** Philip Isett, University of Texas at Austin

**Title:** TBA

**Abstract:** TBA

__15. July 16th, 2020__

*9 pm Beijing Time (GMT+8)*

**Speaker: ** László Székelyhidi Jr**,** Leipzig University

**Title:** TBA

**Abstract:**TBA

*10 pm Beijing Time (GMT+8)*

**Speaker: ** Aaron Naber, Northwestern University

**Title:** TBA

**Abstract:**TBA

*11 pm Beijing Time (GMT+8)*

**Speaker:** Connor Mooney, University of California, Irvine

**Title:** The Bernstein problem for elliptic functionals

**Abstract:** The Bernstein problem asks whether entire minimal graphs in $\mathbb{R}^{n+1}$ are necessarily hyperplanes. This problem was completely solved by the late 1960s in combined works of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti. We will discuss the analogue of this problem for more general elliptic functionals, and some recent progress in the case n = 6.

__16. July 23rd, 2020__

*9 pm Beijing Time (GMT+8)*

**Speaker: **Eduard Feireisl, Czech Technical University, Prague

**Title:** TBA

**Abstract:**TBA

*10 pm Beijing Time (GMT+8)*

**Speaker: **

**Title:** TBA

**Abstract:**TBA

*11 pm Beijing Time (GMT+8)*

**Speaker:** Steve Shkoller, University of California, Davis

**Title:** TBA

**Abstract:** TBA

__17. July 30th, 2020__

*9 pm Beijing Time (GMT+8)*

**Speaker: **Toumo Kuusi, University of Helsinki

**Title:** TBA

**Abstract:**TBA

*10 pm Beijing Time (GMT+8)*

**Speaker: ** Yao Yao, Georgia Institute of Technology

**Title:** TBA

**Abstract:**TBA

*11 pm Beijing Time (GMT+8)*

**Speaker:** Javier Gómez-Serrano, Princeton University

**Title:** TBA

**Abstract:** TBA