A Conference on PDEs

Abstracts

• Jacob Bedrossian — Vortex Axisymmetrization, Inviscid Damping, and Vorticity Depletion in the Linearized 2D Euler Equations

  Coherent vortices are often observed to persist for long times in turbulent 2D flows even at very high Reynolds numbers and are observed in experiments and computer simulations to potentially be asymptotically stable in a weak sense for the 2D Euler equations. We consider the incompressible 2D Euler equations linearized around a radially symmetric, strictly monotone decreasing vorticity distribution. For sufficiently regular data, we prove the inviscid damping of the θ-dependent radial and angular velocity fields with the optimal rates. We moreover prove that the vorticity weakly converges back to radial symmetry as t→∞, a phenomenon known as vortex axisymmetrization in the physics literature, and characterize the dynamics in higher Sobolev spaces. Furthermore, we prove that the θ-dependent angular Fourier modes in the vorticity are ejected from the origin as time goes to infinity, resulting in faster inviscid damping rates than those possible with passive scalar evolution. This non-local effect is called vorticity depletion; our work appears to be the first to find it relevant for the dynamics of vortices. Joint work with Michele Coti Zelati and Vlad Vicol.

• Demetrios Christodoulou — The Shock Development Problem

  Abstract: The study of the physical evolution of a compressible fluid beyond the point where surfaces in spacetime appear where the derivatives of the physical variables blow up leads to the shock development problem. This is a free boundary problem for a quasilinear hyperbolic system of partial differential equations, with nonlinear conditions at the free boundary, the shock hypersurface, and with initial conditions on a transversal characteristic hypersurface which are singular at its past boundary, which is at the same time the past boundary of the shock hypersurface.

  My recent work solves this problem by introducing two new geometric methods and one new analytic method. The first geometric method transforms the problem into a problem of two coupled differential systems on a manifold, the first of which is fully nonlinear. What is achieved by this transformation is a complete regularization of the problem. The second geometric method is concerned with the derivation of energy identities for free boundary problems and features vectorfields in the role of variation fields. The analytic method is designed to overcome the difficulties arising from the severely singular integrals appearing as error integrals in the energy identities.
  

• Charles Collot — Shock Formation for Burgers Equation with Transversal Viscosity

  This talk is about singularity formation for solutions to

  (*) @_tu + u@_xu + @_yyu = 0; (x; y) ∈ R²

  which is a simplified model of Prandtl’s boundary layer equation. Note that it reduces to Burgers equation for y-independent solutions u(t; x; y) = v(t; x). We will first recast the well-known shock formation theory for Burgers equation using the framework of self-similar blow-up. This will provide us with an analytic framework to study the effect of the transversal viscosity. The main result is the construction and precise description of singular solutions to (*). They concentrate a self-similar profile of Burgers equation along the streamwise x variable, whose scaling parameters depend on the transversal direction y and evolve according to a parabolic system. This is joint work with T.E. Ghoul and N. Masmoudi.

• Anne-laure Dalibart — Models for Fluid Boundary Layers: Beyond the Prandtl Equation?

  The Prandtl equation was derived in 1904 by Ludwig Prandtl in order to describe the behavior of fluids with small viscosity around a solid obstacle. Over the past decades, several results of ill-posedness in Sobolev spaces have been proved for this equation. As a consequence, it is natural to look for more sophisticated boundary layer models, that describe the coupling with the outer Euler flow at a higher order. Unfortunately, these models do not always display better mathematical properties, as I will explain in this talk. This is a joint work with Helge Dietert, David Gérard-Varet and Frédéric Marbach.

• Tarek Elgindi — Singularity Formation in Incompressible Fluids

  Abstract:  We will discuss some recent results on finite-time singularity formation for strong solutions to the 3D Euler system and related models. Our analysis is based on scale-invariant solutions and a careful analysis of the 3D Euler system in critical regularity spaces. This is a joint work with I. Jeong.
  

• Patrick Gérard — The Energy Critical Half Wave Maps Equation

  I will discuss the Hamiltonian evolution of maps from the line to the two-dimensional sphere, according to the energy
  

  $$E(u)=\int_{\mathbb R^2}\frac{|u(x)-u(y)|^2}{(x-y)^2}\, dx\, dy$$
  

  In particular, I will show why this Hamiltonian system admits a Lax pair structure, and I will stress  some consequences of this on  qualitative properties of solutions.
  

  This is a joint work with Enno Lenzmann (Basel).
  

• Bob Hardt — A Linear Isoperimetric Inequality for Algebraic and Analytic Varieties

  Abstract: In R^n, the classical n dimensional isoperimetric inequality bounds the volume of any smooth region U by c_n [ H^{n-1}(Bdry U)]^n/(n-1) , where c_n is the constant giving equality with U being an n ball. In higher codimension, any integral k-1 cycle B in R^n with k < n , is the boundary of some k chain T with mass(T) \leq c_k mass(B)^ k/(k-1) . Inside an ambient space X like a complete Riemannian manifold, polyhedral complex, or more generally Euclidean Lipschitz neighborhood retract, every integral boundary bounds some chain satisfying such an isoperimetric inequality with constant C_X depending on X. This relation may fail for singular spaces X with cusps like algebraic varieties With T. DePauw we prove a linear isoperimetric inequality valid for cycles of any dimension in a compact set  X  defined by polynomial or real analytic equalities or inequalities. This generalizes earlier work on codimension one boundaries by L.Bos-P.Milman and B.Hua-FH.Lin.
  

• Slim Ibrahim — Uniqueness of Ground States to Nonlinear Scalar Field Equations Involving the Sobolev Critical Exponent in their Nonlinearities​

  Abstract will be posted when available.

• Alexandru Ionesco — On the Global Regularity of the Einstein-Klein-Gordon System

  Abstract: I will discuss some recent work on questions concerning global regularity for a class of physical evolution equations. I will mostly focus on two-speed evolution problems, such as the Einstein-Klein-Gordon system of General Relativity. This is joint work with Benoit Pausader.
  

• Zhen Lei — Sharp One Component Regularity for Navier-Stokes

  Abstract: In this talk we will first present that one component of velocity has an a priori bound which is critical with respect to the natural scaling of Navier-Stokes in the axi-symmetric case. We then turn to present the study on one component regularity for Navier-Stokes equations. Certain criticality property of the axi-symmetric Navier-Stokes equations will also be reported. 
  

• Ali Maalaoui — Commutator Estimates for the Fractional sub-Laplacian on Carnot Group and Applications

  Abstract: In this talk we discuss a Leibniz rule type estimate involving the fractional sub-Laplacian. That is we provide an point wise estimate to the quantity
  $$H_{\alpha}(u,v)=(-\Delta_{b})^{\frac{\alpha}{2}}(uv)-u(-\Delta_{b})^{\frac{\alpha}{2}}v-v(-\Delta_{b})^{\frac{\alpha}{2}}u.$$
  these type of estimates are fundamental in overcoming the non-local behavior of the operator in order to provide regularity estimates or to study bubbling phenomena. As an application, we provide the characterization of the Palais-Smale sequences for the fractional CR-Yamabe problem on the Heisenberg group and the CR-Sphere.
  

• Van Tien Nguyen — Multiple Collapses for the Two Dimensional Parabolic-Elliptic Keller-Segel System

  

• Fabio Pusateri — On the Water Waves Problem

  Abstract: Following the breakthrough of Germain-Masmoudi-Shatah who proved global existence of small 3D gravity waves, several major results have been proven in recent years about the global regularity for solutions of the Water Waves equations. I will describe the model and some of these main results, and in particular joint works with A. Ionescu on 2D water waves and with Y. Deng, A. Ionescu, and B. Pausader on the full gravity-capillary 3D problem. I will also discuss a recent result on the extended lifespan of multi-dimensional periodic gravity-capillary water waves.
  

• Tristan Rivière — The Regularity of Parametrized Stationary Varifold and Applications to the Variational Theory of Minimal Surfaces

  The regularity of parametrized stationary varifold and applications to the variational theory of minimal surfaces.
  

• Belkacem Said-Houari — Global Existence and Decay Estimates for the Solution of a Nonlinear Bresse System

  Abstract: Our main focus in this talk is to study the global existence and asymptotic behavior of a nonlinear version of the Bresse system. We show first that the linear version of this system is of regularity-loss type. This regularity loss creates difficulties when dealing with the nonlinear problem since the dissipative property of the problem becomes very weak in the high frequency region and as a result the classical energy method fails. To overcome this difficulty, we use an energy method with negative weights to create an articial damping which allows us to control the nonlinearity.

• Mickael Struwe — Harmonic Spheres

  Abstract: By results of El Soufi, harmonic maps from spheres to a closed target are saddle points of the energy with respect to the action of the Moebius group on the domain. We use this to devise a strategy for finding nontrivial harmonic maps from spheres of dimension 3 and higher.
  

• Edriss Titi — Analysis of Certain Geophysical Models: Global Regularity and Finite-time Blowup Results

  Abstract: In this talk I will present some recent results concerning global regularity of certain geophysical models. This will include the three-dimensional primitive equations with various anisotropic viscosity and turbulence mixing diffusion, and certain tropical atmospheric models with moisture. Moreover, I will show that in the non-viscous (inviscid) case there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup).
  

• Amjad Tufaha — On Local-in-Time Existence of Solutions to the Free Moving Boundary Incompressible Euler Equations Without Surface Tension

  Abstract: We establish a priori estimates for local-in-time existence of solutions to the water wave model consisting of the 3D incompressible Euler equations on a domain with a free surface, without surface tension, under minimal regularity assumptions on the initial data and the Rayleigh-Taylor sign condition. The initial data is allowed to be rotational and it is assumed to belong to Sobolev space H2.5+δH2.5δ, where δ>0δ0 is arbitrary. This is joint work with Igor Kukavica from the University of Southern California.
  

• Junchen Wei — Superfluids Passing an Obstacle, Gross-Pitaevskii, and KP-I

  Abstract: We consider  a superfluid described by the Gross-Pitaevskii equation passing an obstacle
  $$ \epsilon^2 \Delta u + u(1-|u|^2)=0 \ \mbox{in}\  R^d \backslash \Omega$$

  We prove the existence of two type of solutions: vortex free, and single vortex solution. We show a deep connection with the traveling wave equation of Gross-Pitaevskii

  $$ ic \frac{\partial u}{\partial y} +\Delta u+u     (1-|u|^2)=0 \mbox{in} \ \R^2 $$

  We discuss new solutions found for the traveling wave equation when the speed $c$ is either small or close to sound speed $ \sqrt{2}$. Connection with KP-I

  $$ \partial_{x}^{2}\left(  \partial_{x}^{2}u-u+3 u^2 \right) -\partial_{y}^{2}u=0.
  $$ and  nondegeneracy  of (single and multiple) lump solutions will be discussed.  

  Joint work with FH Lin, Y. Liu
  

• Hatem Zaag — Blow-Up Solutions for Two Non-Variational Semilinear Parabolic Systems

  We consider two non-variational semilinear parabolic systems, with different diffusion constants between the two components. The reaction terms are of power type in the first system. They are of exponential type in the second. Using a formal approach, we derive blow-up profiles for those systems. Then, linearizing around those profiles, we give the rigorous proof, which relies on the two-step classical method: (i) the reduction of the problem to a finite-dimensional one, then, (ii) the proof of the latter thanks to Brouwer's lemma.

  In comparison with the standard semilinear heat equation, several technical problems arise here, and new ideas are needed to overcome them.

  This is a joint work with T. Ghoul and V.T. Nguyen from NYU Abu Dhabi.
  

• Day One — Sunday, January 14, 2018

  Day One Agenda

  Time	Event
  8:30-9am	Registration Institute Conference Center (A6), Meeting Room Foyer
  9-9:10am	Welcome Remarks and Indroduction
  9:10-10:00am	Demetrios Christodoulou, ETH Zurich
  10-10:30am	Break and Refreshments Meeting Room Foyer
  10:30-11:20am	C. Mouhot, University of Cambridge
  11:30am-12:20pm	J. Bedrossian, University of Maryland
  12:20-2pm	Lunch Atrium
  2-2:50pm	F. Pusateri, University of Princeton
  2:55-3:45pm	B. Said Houari, University of Sharjah
  3:45-4:15pm	Break and Refreshments Meeting Room Foyer
• Day Two — Monday, January 15, 2018

  Day Two Agenda

  Time	Event
  9-10am	E. Titi, Texas A&M
  10-10:30am	Break and Refreshments Meeting Room Foyer
  10:30-11:20am	T. Elgindi, University of San Diego
  11:30am-12:20pm	Z. Lei, Fudan University
  12:20-2pm	Lunch Atrium
  2-2:50pm	A.L. Dalibart, University of Paris VI
  2:55-3:45pm	V.T. Nguyen, NYUAD
  3:44-4:15pm	Break and Refreshments Atrium
  6:30-8pm	Public Program and Reception: From Differential Equations to Data Science and Back  Conference Center (A6)
• Day Three — Tuesday, January 16, 2018

  Day Three Agenda

  Time	Event
  8:30-9:20am	J.C. Wei, University of British Columbia
  9:30-10:20am	S. Ibrahim, University of Victoria
  10:20-10:50am	Break and Refreshments Meeting Room Foyer
  10:50-11:40am	R. Caflisch, NYU New York
  11:50am-12:40pm	C. Collot, NYUAD
  12:40-2pm	Lunch Atrium
• Day Four — Wednesday, January 17, 2018

  Day Four Agenda

  Time	Event
  8:30-9am	T. Riviere, ETH Zurich; Special Session about F. Lin
  9:10-10am	C. Delellis, ETH Zurich
  10-10:30am	Break and Refreshments Meeting Room Foyer
  10:30-11:20am	T. Riviere, ETH Zurich
  11:30am-12:20pm	B. Hardt, Rice University
  12:20-2pm	Lunch Atrium
  2-2:50pm	H. Zaag, University of Paris XIII
  2:55-3:45pm	A. Ionescu, Princeton University
  3:45-4:15pm	Break and Refreshments Meeting Room Foyer
  4:15-5:05pm	Amjad Tuffaha, American University of Sharjah
• Day Five — Thursday, January 18, 2018

  Day Five Agenda

  Time	Event
  8:30-9:20am	M. Struwe, ETH Zurich, Special Session About J. Shatah
  9:30-10:20am	Y. Deng, NYU New York
  10:20-10:50am	Break and Refreshments Meeting Room Foyer
  10:50-11:40am	P. Gerard, University of Paris XI
  11:50am-12:40am	A. Maalaoui, AURAK
  12:40-2pm	Lunch Atrium
  2-2:50pm	C. Zheng, Georgia Tech
  2:55-3:45pm	M. Struwe, ETH Zurich
  3:45-4:15pm	Break and Refreshments Meeting Room Foyer
  4:15-5:05	S. Ulusoy, AURAK