# Nonlinear Analysis Seminar ---- Spring 2021 (Organizers: Yanyan Li, Dennis Kriventsov)

• The seminar is run remotely on Zoom mostly on Wednesdays (9:30am--10:30am).

Zoom Link   Meeting ID: 941 7571 5705   Passcode: 849396

• May 5, 2021, Wed, 9:30-10:30am (Eastern Time)

Speaker: Ian Tobasco, University of Illinois at Chicago

Title: Exact solutions for the wrinkle patterns of confined elastic shells

Abstract: A basic fact of geometry is that there are no length-preserving maps from a sphere to the plane. But what happens if you confine a thin elastic shell, which prefers to be a curved surface but can deform approximately isometrically, to reside nearby a plane? It wrinkles, and forms a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematical modeling of thin elastic shells, this talk will focus on a new set of simple, geometric rules we have derived for wrinkle patterns via Gamma-convergence and convex analysis of the limit problem. Our rules govern the asymptotic layout of the wrinkle peaks and troughs --- for instance, negatively curved wrinkles tend to arrange along segments solving the minimum exit time problem, in the infinitesimally wrinkled limit. Positively curved shells can be understood more or less completely as well, through a hidden duality with their negatively curved counterparts. Our predictions for the wrinkle patterns of confined shells match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.

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• Feb. 3, 2021, Wed, 9:30--10:30am

Speaker: Camillo De Lellis, Institute for Advanced Study

Title: Flows of vector fields: classical and modern

Abstract: Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindel\"of) Theorem states that, if the vector field $v$ is Lipschitz in space, for every initial datum $x$ there is a unique trajectory $\gamma$ starting at $x$ at time $0$ and solving the ODE $\dot{\gamma} (t) = v (t, \gamma (t))$. The theorem looses its validity as soon as $v$ is slightly less regular. However, if we bundle all trajectories into a global map allowing $x$ to vary, a celebrated theory put forward by DiPerna and Lions in the 80es show that there is a unique such flow under very reasonable conditions and for much less regular vector fields. A long-standing open question is whether this theory is the byproduct of a stronger classical result which ensures the uniqueness of trajectories for {\em almost every} initial datum. I will give a complete answer to the latter question and draw connections with partial differential equations, harmonic analysis, probability theory and Gromov's $h$-principle.

• Feb. 10, 2021, Wed, 9:30--10:30am

Speaker: Hector Chang-Lara, CIMAT (Centro de Investigacion en Matematicas A.C.)

Title: Eikonal vs. Brownian: Regularity for the solution of an equation with gradient constraint

Abstract: Two controllers are in charge of steering a spaceship in some domain Omega. The first controller wants to spend as much time as possible exploring Omega while the second wants to get out of it as quickly as possible. The first controller determines minute by minute whether the ship is moving by a Brownian motion or with constant speed, in which case it is the second controller who chooses the direction. Under these instructions, determining the optimal strategies for each player leads us to solve the equation min (-Delta u, |Du|) = 1 which has several interesting characteristics. Among them is the presence of a free boundary which separates the regions where a Poisson or an Eikonal equation is satisfied. In a recent collaboration with Edgard Pimentel (PUC-Rio) we showed that the solutions are Lipschitz continuous and that |Du| is continuous, even though the gradient is discontinuous in numerous examples. This problem is a simplification of interesting models in financial mathematics related with the optimal strategy for the payment of dividends from multiple insurances.

• Feb. 17, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Juncheng Wei, The University of British Columbia

Title: Recent results on Allen-Cahn equation

Abstract: I will discuss some recent results on Allen-Cahn equation, including the half space theorem and stability of saddle solutions in R^{8,10,12}.

• Feb. 24, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Hui Yu, Columbia University

Title: The N-membrane problem

Abstract: The N-membrane problem is the study of shapes of elastic membranes being pushed against each other. The main questions are the regularity of the functions modeling the membranes, and the regularity of the contact regions between consecutive membranes. These are classical questions in free boundary problems. However, very little is known when N is larger than 2. In this case, there are multiple free boundaries that cross each other, and most known techniques fail to apply. In this talk, we discuss, for general N, the optimal regularity of the solutions in arbitrary dimensions, and a classification of blow-up solutions in 2D. Then we focus on the regularity of the free boundaries when N=3. We also discuss how the techniques developed here can be applied to other problems involving multiple free boundaries. This talk is based on two recent joint works with Ovidiu Savin (Columbia University).

• March 3, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Monica Musso, University of Bath

Title: Desingularization of vortices and leapfrogging phenomena for Euler equations

Abstract: In 1858, Helmholtz predicted that two vortex-rings in fluids will pass through each other and become leapfrogging. In this talk I will give a rigorous justification of Helmholtz’s conjecture for 3D Euler flow, and also discuss the dynamical desingularization of vortices for 2D Euler flows. The key ingredients are new gluing methods for Euler flows. (Joint work with Juan Davila, Manuel del Pino and Juncheng Wei.)

• March 10, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Francois Hamel, Aix-Marseille University

Title: Symmetry properties for the Euler equations and semilinear elliptic equations

Abstract: In this talk, I will discuss radial and one-dimensional symmetry properties for the stationary incompressible Euler equations in dimension 2 and some related semilinear elliptic equations. I will show that a steady flow of an ideal incompressible fluid with no stagnation point and tangential boundary conditions in an annulus is a circular flow. The same conclusion holds in complements of disks as well as in punctured disks and in the punctured plane, with some suitable conditions at infinity or at the origin. I will also discuss the case of parallel flows in two-dimensional strips, in the half-plane and in the whole plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on radial and one-dimensional symmetry results for the solutions of some elliptic equations satisfied by the stream function. The talk is based on some joint works with N. Nadirashvili.

• March 24, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Manuel del Pino, University of Bath

Title: Singularity formation for the Keller-Segel system in the plane

Abstract: The classical model for chemotaxis is the planar Keller-Segel system $$u_t = \Delta u - \nabla\cdot ( u\nabla v ), \quad v(\cdot, t) = \frac 1{2\pi} \log 1{|\cdot |} * u(\cdot ,t) .$$ in $\R^2\times (0,\infty)$. Blow-up of finite mass solutions is expected to take place by aggregation, which is a concentration of bubbling type, common to many geometric flows. We build with precise profiles solutions in the critical-mass case $8\pi$, in which blow-up in infinite time takes place. We establish stability of the phenomenon detected under arbitrary mass-preserving small perturbations and present new constructions in the finite time blow-up scenario.

• March 31, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Jeff Calder, Universoty of Minnesota

Title: PDE continuum limits for prediction with expert advice

Abstract: Prediction with expert advice refers to a class of machine learning problems that is concerned with how to optimally combine advice from multiple experts whose prediction qualities may vary greatly. We study a stock prediction problem with history-dependent experts and an adversarial (worst-case) market, posing the problem as a repeated two-player game. The game is a generalization of the Kohn-Serfaty two-player game for curvature motion. We prove that when the game is played for a long time, the discrete value functions converge in the continuum to the solution of a nonlinear parabolic partial differential equation (PDE) and the optimal strategies are characterized by the solution of a Poisson equation over a De Bruijn graph, arising from the history-dependence of the experts. Joint work with Nadejda Drenska (UMN).

• April 7, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Siyuan Lu, McMaster University

Title: Rigidity of Riemannian Penrose inequality with corners and its implications

Abstract: Motivated by the rigidity case in the localized Riemannian Penrose inequality, we show that suitable singular metrics attaining the optimal value in the Riemannian Penrose inequality is necessarily smooth in properly specified coordinates. If applied to hypersurfaces enclosing the horizon in a spatial Schwarzschild manifold, the result gives the rigidity of isometric hypersurfaces with the same mean curvature. This is a joint work with Pengzi Miao.

• April 14, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Qi S. Zhang, University of California Riverside

Title: Time analyticity and reversibility of some parabolic equations

Abstract: We describe a concise way to prove time analyticity for solutions of parabolic equations including the heat and Navier Stokes equations. In some cases, results under sharp conditions are obtained. An application is a necessary and sufficient condition for the solvability of the backward heat equation which is ill-posed, helping to remove an old obstacle in control theory. Part of the work is joint with Hongjie Dong.

• April 21, 2021, Wed, 9:30--10:30am (Eastern Time)

Speaker: Qing Han, University of Notre Dame

Title: Solutions of the Minimal Surface Equation and of the Monge-Ampere Equation near Infinity

Abstract: Classical results assert that, under appropriate assumptions, solutions near infinity are asymptotic to linear functions for the minimal surface equation (due to Bers and Schoen) and to quadratic polynomials for the Monge-Ampere equation (due to Caffarelli-Li) for dimension n at least 3, with an extra logarithmic term for n=2. We characterize remainders in the asymptotic expansions by a single function, which is given by a solution of some elliptic equation near the origin via the Kelvin transform. Such a function is smooth in the entire neighborhood of the origin for the minimal surface equation in every dimension and for the Monge-Ampere equation in even dimension, but only C^{n-1,\alpha} for the Monge-Ampere equation in odd dimension, for any \alpha in (0,1).

• April 27, 2021, Tuesday, 1:40-2:40pm (Eastern Time)

Speaker: Thomas Yizhao Hou, California Institute of Technology

Title: The Interplay between Analysis and Computation in Studying 3D Euler Singularity

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 and the original De Gregorio model develop finite time self-similar singularity. This analysis has been generalized to prove finite time singularity of the 2D Boussinesq and 3D Euler equations with C^{1,\alpha} initial velocity and boundary, whose solutions share some essential features similar to those reported in the Luo-Hou computation. Finally, we present some recent numerical results on singularity formation of the 3D axisymmetric Navier-Stokes equations with degenerate diffusion coefficients.

# Nonlinear Analysis Seminar ---- Fall 2020 (Organizers: Yanyan Li, Dennis Kriventsov)

• The seminar is run remotely on Zoom mostly on Wednesdays (9:30am--10:30am).

Registration Link   Meeting ID: 992 8122 5008

• September 16, 2020, Wed, 9:30--10:30am

Speaker: Luis Caffarelli, University of Texas at Austin

Title: Propagation on a domain, driven by expansion on its boundary

Abstract: Diverse models of propagation of a species that travels along a line and diffuses sideways were introduced among others by Berestycki, Coulon, Roquejoffre, Rossi I will describe work in collaboration with Roquejoffre and Tomasetti in this area of research.

• September 23, 2020, Wed, 9:30--10:30am

Speaker: Hao Jia, University of Minnesota

Title: Long time dynamics of 2d Euler and nonlinear inviscid damping

Abstract: In this talk, we will discuss some joint work with Alexandru Ionescu on the nonlinear inviscid damping near point vortex and monotone shear flows in a finite channel. We will put these results in the context of long time behavior of 2d Euler equations and indicate further important open problems in the field.

• September 30, 2020, Wed, 9:30--10:30am

Speaker: Yao Yao, Georgia Institute of Technology

Title: Two results on the interaction energy

Abstract: For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] \le E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities. I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m\ge 2, where the threshold is sharp.

• October 7, 2020, Wed, 9:30--10:30am

Speaker: Alexandru Ionescu, Princeton University

Title: On the global regularity for the Einstein-Klein-Gordon coupled system

Abstract: In joint work with Benoit Pausader we consider the Einstein field equations of General Relativity for self-gravitating massive scalar fields (the Einstein-Klein-Gordon system). Our main results concern the global regularity, modified scattering, and precise asymptotic analysis of solutions of this system with initial data in a small neighborhood of the Minkowski space-time.

• October 14, 2020, Wed, 9:30--10:30am

Speaker: Weiyong He, University of Oregon

Title: Harmonic and biharmonic almost complex structures

Abstract: Harmonic almost complex structures was introduced by C. Wood in 1990s. We study the existence and regularity of weak harmonic almost complex structure from analytic point of view, as a tensor-valued version of harmonic maps. We also introduce the notion of biharmonic almost complex structures, in particular in dimension four. We prove that biharmonic almost complex structures are smooth, and there always exist an energy-minimizing biharmonic almost complex structure. Moreover, given a homotopy class, we also show existence results. We conjecture that the homotopy class of energy-minimizing biharmonic almost complex structures do not depend on a generic background metric.

• October 21, 2020, Wed, 9:30--10:30am

Speaker: Yannick Sire, Johns Hopkins University

Title: Blow-up solutions via parabolic gluing

Abstract: We will present some recent results on the construction of blow-up solutions for critical parabolic problems of geometric flavor. Initiated in the recent years, the inner/outer parabolic gluing is a very versatile parabolic version of the well-known Lyapunov-Schmidt reduction in elliptic PDE theory. The method allows to prove rigorously some formal matching asymptotics (if any available) for several PDEs arising in porous media, geometric flows, etc….I will give an overview of the strategy and will present several applications to (variations of) the harmonic map flow, Yamabe flow and Yang-Mills flow. I will also present some open questions.

• October 28, 2020, Wed, 9:30--10:30am

Speaker: Zhifei Zhang, Peking University

Title: Linear stability of pipe Poiseuille flow at high Reynolds number regime

Abstract: The linear stability of pipe Poiseuille flow is a long standing problem since Reynolds experiment in 1883. Joint with Qi Chen and Dongyi Wei, we solve this problem at high Reynolds regime. We first introduce a new formulation for the linearized 3-D Navier-Stokes equations around this flow. Then we establish the resolvent estimates of this new system under favorable artificial boundary conditions. Finally, we solve the original system by constructing a boundary layer corrector.

• November 4, 2020, Wed, 9:30--10:30am

Speaker: Daniela De Silva, Barnard College

Title: On the Boundary Harnack Principle

Abstract: In this talk we discuss the classical Boundary Harnack Principle and some of its applications to Free Boundary problems. We then present a recent direct analytic proof of this result, for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.

• November 11, 2020, Wed, 9:30--10:30am

Speaker: Fanghua Lin, Courant Institute

Title: Defects in heat flows of harmonic maps

Abstract: Starting with Struve's solution of the heat flow harmonic maps from a surface, we shall discuss several model heat flow problems for maps and its associated singularities. In case there are time or energy scale separations, one can often derive dynamical laws for singularities (energy concentration sets). There are challenging open problems when such scale separation don't exist such as defect motion in 3D liquid crystal evolutions or coupled flows of sharp interfaces and phase-valued maps in models of fast reaction--slow diffusion described by Keller-Rubinstein-Steinberg.

• November 18, 2020, Wed, 9:30--10:30am

Speaker: Alessio Figalli, ETH

Title: Generic regularity in obstacle problems

Abstract: The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and which is constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. In a recent paper with Ros-Oton and Serra we show that, generically, the singular set has zero $\mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary), solving a conjecture of Schaeffer in dimension $n \leq 4$. The aim of this talk is to give an overview of these results.

• November 25, 2020, Wed, 9:30--10:30am

Speaker: Shaoming Guo, University of Wisconsin

Title: Fourier restrictions and decouplings for surfaces of co-dimension two

Abstract: I will report some recent process on Fourier restriction estimates and decoupling inequalities for quadratic surfaces of co-dimension two.

• December 2, 2020, Wed, 9:30--10:30am

Speaker: Maria Esteban, Universite Paris-Dauphine

Title: Flows and symmetry properties of positive solutions of nonlinear elliptic PDEs

Abstract: In this talk will be presented various results concerning the symmetry and symmetry breaking properties of positive solutions of nonlinear elliptic PDEs. In some cases related to the study of Caffarelli-Kohn-Nirenberg inequalities the use of flows can provide optimal results concerning symmetry or symmetry breaking. In other cases, where magnetic Laplacians are concerned, the situation becomes more complicated, and one cannot always prove optimal results. Various examples will be discussed during the talk.

• December 9, 2020, Wed, 9:30--10:30am

Speaker: Angela Pistoia, Universita' Degli Studi Di Roma La Sapienza.

Title: Elliptic systems with critical growth

Abstract: I will present some results concerning existence and phase separation of entire solutions to a pure critical competitive elliptic system and their relation with sign-changing solutions to the Yamabe problem.

• December 16, 2020, Wed, 9:30--10:30am

Speaker: Xinan Ma, University of Science and Technology of China

Title: Liouville theorem for a class of semilinerar elliptic problems on Heisenberg group

Abstract: We obtain an entire Liouville type theorem to the classical semilinear subcritical elliptic equation on Heisenberg group. A pointwise estimate near the isolated singularity was also proved. The soul of the proofs is an a priori integral estimate, which deduced from a generalized formula of that found by Jerison and Lee. This is a joint work with Prof. Qianzhong Ou.