Program
Long Time Behavior and Singularity Formation in PDEs  Part II
Sunday, January 10, 2021
All times are in Gulf Standard Time (GST)

Time: 4:30  5:30pm
Speaker: Jeanmarc Delort

Time: 8  9pm
Speaker:Sasha Kiselev
Abstract
The question of how roots of polynomials move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. Interestingly, Dimitri Shlyakhtenko and Terry Tao have later formally obtained the same PDE as the evolution equation for free fractional convolution of a measure, an object appearing in free probability. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species  such as fish, birds or ants. The equation is critical, but due to strongly nonlinear form of its coefficients, proving global regularity for its solutions is harder than for equations such as Burgers, SQG or Euler alignment model. I will discuss joint work with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.

Time: 8  9pm
Speaker: Andrea Nahmod
Abstract
In this talk we focus on the time dynamics of solutions of periodic nonlinear Schrödinger with random initial data and the key underlying difficulty of understanding how randomness propagates under the nonlinear flow. We discuss the theory of random tensors, a powerful new framework that we developed with Yu Deng and Haitian Yue.
4:20pm Opening remarks, by Arlie Petters, NYUAD Provost
Monday, January 11, 2021
All times are in Gulf Standard Time (GST)

Time: 4:30  5:30pm
Speaker: Belkacem SaidHouari, Associate Professor, Mathematics, University of Sharjah
Abstract
We consider the Cauchy problem of a thirdorder in time nonlinear equation known as the Jordan—MooreGibsonThompson equation arising in acoustics as an alternative model to the wellknown Kuznetsov equation. We show a local existence result in appropriate function spaces, and using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, we obtain polynomial decay rates in time for a norm related to the solution. 
Time: 8  9pm
Speaker: Luis Vega, BCAM UPV/EHU Research Professor, Group Leader, La Universidad del País Vasco
Abstract
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a nonobvious non linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.

Time: 9  10pm
Speaker: Emmanuel Grenier
Abstract
In this talk we review recent results on the instability of shear layers for the incompressible Navier Stokes equations as the viscosity goes to 0.
Tuesday, January 12, 2021
All times are in Gulf Standard Time (GST)

Time: 2  3pm
Speaker: Boualem Djehiche, Professor of mathematical statistics, KTH Royal Institute of Technology
Abstract
Boualem will review some recent results on a problem of Mass Transport of Particles Towards a Target in a Brownian environment.

Time: 8  9pm
Speaker: Vlad Vicol, Professor of Mathematics, Courant Institute of Mathematical Sciences, New York University
Abstract
We discuss the formation of singularities (shocks) for the compressible Euler equations with the ideal gas law. We provide a constructive proof of stable shock formation from smooth initial datum, of finite energy, and with no vacuum regions. Additionally, for the nonisentropic problem, we show that sounds waves interact with entropy waves to produce vorticity at the shock. This talk is based on joint work with Tristan Buckmaster and Steve Shkoller.

Time: 9  10pm
Speaker: Edriss S. Titi, Arthur Owen Professor of Mathematics, Texas A&M University
Abstract
Large scale dynamics of the oceans and the atmosphere is governed by the primitive equations (PEs). It is wellknown that the threedimensional viscous primitive equations are globally wellposed in Sobolev spaces. In this talk, I will discuss the illposedness in Sobolev spaces, the local wellposedness in the space of analytic functions, and the finitetime blowup of solutions to the threedimensional inviscid PEs with rotation (Coriolis force). Eventually, I will also show, in the case of “wellprepared" analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the lifespan of the solutions which grows toward infinity with the rotation rate. The latter is achieved by a delicate analysis of a simple limit resonant system whose solution approximates the corresponding solution of the 3D inviscid PEs with the same initial data.
Wednesday, January 13, 2021
All times are in Gulf Standard Time (GST)

Time: 4:30  5:30pm
Speaker: Nejla Nouaili, Assistant Professor of Paris Dauphine University, Paris (UPD)
Abstract
In this talk Nejla will present a construction of a solution for the Complex GinzburgLandau (CGL) equation in a general critical case, which blows up in finite time T only at one blowup point. She will also give a sharp description of its profile. These is joint work with G.K.Duong and H.Zaag. 
Time: 8  9pm
Speaker: Massimiliano Berti, Full professor of Mathematical Analysis at SISSA
Abstract
In this talk Massimiliano will present some new results about bifurcation of time quasiperiodic solutions for fluid equations

Time: 9  10pm
Speaker: Hajer Bahouri, Research Director of the National Center for Scientific Research at the University ParisEstCréteilValdeMarne
Abstract
In this work, in collaboration with Galina Perelman, we prove that the derivative nonlinear Schrödinger equation is globally wellposed for general Cauchy data in $H^{\frac 1 2}(\R)$. The proof of our result is achieved by combining the profile decomposition techniques with the integrability structure of the equation.
Thursday, January 14, 2021
All times are in Gulf Standard Time (GST)

Time: 4:30  5:30pm
Speaker: AnneLaure Dalibard, Professor at Sorbonne Université
Abstract
The purpose of this talk is to present two 1d congestion models: a « soft » congestion model in which the congestion effects are described by a singular pressure law, and a « hard » congestion model in which the dynamics are different in congested zones (incompressible dynamics) and in the noncongested ones (compressible dynamics). For each model, we exhibit traveling fronts and investigate their stability. This is a joint work with Charlotte Perrin. 
Time: 8  9pm
Speaker: Jose Carrillo, Professor of the Analysis of Nonlinear Partial Differential Equations
Abstract
The main goal of this talk is to discuss the stateoftheart in understanding the phenomena of phase transitions for a range of nonlinear FokkerPlanck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as CuckerSmale models for consensus, the Keller Segel model for chemotaxis, and the Kuramoto model for synchronization. We will show the existence of phase transitions in a variety of these models using the natural free energy of the system and their interpretation as natural gradient flow structure with respect to the Wasserstein distance in probability measures. We will discuss both theoretical aspects as well as numerical schemes and simulations keeping those properties at the discrete level.

Time: 9  10pm
Speaker: Charles Collot