Join the weekly PDEA Webinar!
The Covid-19 pandemic has led to the cancellation of numerous conferences and seminars around the world, which clearly has a negative impact on traditional forms of scientific knowledge exchange. With this Webinar, our goal is to contribute an initiative to fill that gap. Very much in line with the scope of this journal, we would like to "encourage and amplify the transfer of knowledge between scientists with different backgrounds and from different disciplines who study, solve or apply the same types of equations". At the same time, we are taking this opportunity to explore new forms of communication in journal publishing. Presentations in the Webinar usually be based on research published in the journal. This will benefit readers, who might use the Webinar as an opportunity to ask questions or make suggestions in a direct exchange with the authors. Vice versa, viewers of the Webinar might appreciate having the published papers at hand in order to delve deeper into the topics discussed. And finally, authors will benefit from the heightened visibility their research receives.
Webinars take place on Thursdays on a weekly basis at 9am EDT, 3pm CET (click here to convert to your time zone). They are held via Zoom and are free to attend (please register in advance). Registration is now open!
Video recordings of the seminars are uploaded here.
Organising Committee
Habib Ammari (ETH Zurich)
Hyeonbae Kang (Inha University)
Lin Lin (UC Berkely)
Sid Mishra (ETH Zurich)
Stanley Snelson (Florida Tech)
Eduardo Teixeira (University of Central Florida)
Zhi-Qiang Wang (Utah State University)
Zhitao Zhang (Chinese Academy of Sciences)
2021 Webinar Schedule
(all talks take place at 3pm Central European Time)
Date | Speaker | Title | Related PDEA Paper |
08/02 | Huyên Pham (Université Paris Diderot) | Solving mean-field PDEs with symmetric neural networks | |
15/02 | Eitan Tadmor (University of Maryland) | Existence and large time behavior in hydrodynamic swarming video | |
22/02 | Diogo Gomes (KAUST) | Displacement convexity and mean-field games | https://rdcu.be/cdUBr |
04/03 | Markus Melenk (TU Wien) | High order numerical methods for fractional diffusion in polygons video | |
11/03 | Kanishka Perera (Florida Tech) | An abstract critical point theorem with applications to elliptic problems with combined nonlinearities video | |
18/03 | Giuseppe Mingione (Università di Parma) | Nonuniformly elliptic problems | |
25/03 | Mouhamed Moustapha Fall (AIMS-Senegal) | Constant (nonlocal) Mean curvature surfaces video | |
29/04 | Camillo De Lellis (IAS Princeton) | Flows of vector fields: classical and modern | |
06/05 | Ian Tice (CMU) | Traveling wave solutions to the free boundary Navier-Stokes equations slides | |
13/05 | Christian Kühn (TU Munich) | Geometric Singular Perturbation Theory for Fast-Slow PDEs | https://rdcu.be/ci6jv |
20/05 | Hung Vinh Tran (University of Wisconsin Madison) | Large time behavior and large time profile of viscous Hamilton-Jacobi equations slides | |
27/05 | Chi-Wang Shu (Brown University) | Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs | |
03/06 | Anna Mazzucato (Penn State) | Direct and inverse problems for a model of dislocations in geophysics slides | |
10/06 | Philipp Grohs (University of Vienna) | Deep Learning in Numerical Analysis | |
17/06 | Apala Majumdar (University of Strathclyde) | PDE problems in the Landau-de Gennes theory for Nematic Liquid Crystals | |
24/06 | Yoichiro Mori (University of Pennsylvania) | Analysis of the Dynamics of Immersed Elastic Filaments in Stokes Flow |
More talks will be added over the course of time.
2021 Speakers and Abstracts
February 8, 2021 -- Huyên Pham -- Solving mean-field PDEs with symmetric neural networks
We propose numerical methods for solving non-linear partial differential equations (PDEs) in the Wasserstein space of probability measures, which arise notably in the optimal control of mean-field dynamics, and are motivated by models of large population of interacting agents.
The method relies first on the approximation of the PDE in infinite dimension by a backward stochastic differential equation (BSDE) with a forward system of $N$ interacting particles. We provide the rate of convergence of this finite-dimensional probabilistic approximation for the solution to the PDE and its $L$-derivative. Next, by exploiting the symmetry of the particles system, we design a machine learning algorithm based on certain types of neural networks, named PointNet and DeepSet, for computing simultaneously the pair solution to the BSDE by backward induction through sequential minimization of loss functions. We illustrate the efficiency of the PointNet/DeepSet networks compared to classical feedforward ones, and provide some numerical results of our algorithm for the examples of a mean-field systemic risk and a mean-variance problem.
February 15, 2021 -- Eitan Tadmor -- Existence and large time behavior in hydrodynamic swarming
I will discuss recent developments in our study of hydrodynamic swarming driven by symmetric communication kernels. A main question of interest is the emergent behavior which we quantify in terms of the spectral gap of a weighted Laplacian associated with the alignment operator. I will derive threshold conditions which guarantee existence of multiD strong solutions, their large time flocking behavior with long-range kernels, and mention open questions for in the presence of short-range kernels.
February 22, 2021 -- Diogo Gomes -- Displacement convexity and mean-field games
In the theory of mean-field games (MFGs), a priori estimates play a crucial role to prove the existence of classical solutions. In particular, uniform bounds for the density of players' distribution and its inverse are of utmost importance. Here, inspired by previous results in the optimal transport theory, we investigate a priori bounds for a first-order planning problem with a non-vanishing potential and establish a displacement-convexity property. Using Moser's iteration method, we show that if the potential satisfies a certain smallness condition, then a displacement-convexity property holds. This property enables Lq bounds for the density. In the one-dimensional case, the displacement-convexity property also gives Lq bounds for the inverse of the density. Finally, using these Lq estimates and Moser's iteration method, we obtain L1 estimates for the density of the distribution of the players and its inverse. We conclude with an application of our estimates to prove the existence and uniqueness of solutions for a first-order mean-field planning problem.
March 4, 2021 -- Markus Melenk -- High order numerical methods for fractional diffusion in polygons
March 11, 2021 -- Kanishka Perera -- An abstract critical point theorem with applications to elliptic problems with combined nonlinearities
We will present an abstract critical point theorem based on a cohomological index theory that produces pairs of nontrivial critical points with nontrivial higher critical groups. This theorem yields pairs of nontrivial solutions that are neither local minimizers nor of mountain pass type for problems with combined nonlinearities. Applications will be given to subcritical and critical p-Laplacian problems, Kirchhoff type nonlocal problems, and critical fractional p-Laplacian problems.
March 18, 2021 -- Giuseppe Mingione -- Nonuniformly elliptic problems
Regularity for nonuniformly elliptic problems has been developed intensively over the last years, especially due to the interest raised by a few special model examples coming from the Calculus of Variations. I shall first give a brief overview of main results in the variational setting and then I shall present some more recent ones obtained in collaboration with Cristiana De Filippis (Università di Torino).
March 25, 2021 -- Mouhamed Moustapha Fall -- Constant (nonlocal) Mean curvature surfaces
The notion of Nonlocal Mean Curvature (NMC) appears recently in the mathematics literature. Alike their local counterpart, it is an extrinsic geometric quantity that is invariant under global reparameterization of a surface and provide a natural extension of the classical mean curvature. We describe some properties of the NMC and the quasilinear differential operators that are involved when it acts on graphs. We also survey recent results on surfaces having constant NMC and describe their intimate link with some problems arising in dibock-copolymer and the study of overdetermined boundary value problems.
April 29, 2021 -- Camillo De Lellis -- Flows of vector fields: classical and modern
Consider a (possibly time-dependent) vector field $v$ on the Euclidean space. The classical Cauchy-Lipschitz (also named Picard-Lindelöf) 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 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.
May 6, 2021 -- Ian Tice -- Traveling wave solutions to the free boundary Navier-Stokes equations
Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity. This is joint work with Giovanni Leoni.
May 13, 2021 -- Christian Kühn -- Geometric Singular Perturbation Theory for Fast-Slow PDEs
Systems with multiple time scales appear in a wide variety of applications. Yet, their mathematical analysis is challenging already in the context of ODEs, where about four decades were needed to develop a more comprehensive theory based upon invariant manifolds, desingularization, variational equations, and many other techniques.
Yet, for PDEs progress has been extremely slow due to many obstacles in generalizing several ODE methods. In my talk, I shall report on two recent advances for fast-slow PDEs, namely the extension of slow manifold theory for unbounded operators driving the slow variables, and the design of a blow-up method for PDEs, where normal hyperbolicity is lost. This is joint work with Maximilian Engel and Felix Hummel.
May 20, 2021 -- Hung Vinh Tran -- Large time behavior and large time profile of viscous Hamilton-Jacobi equations
I will describe our recent results on large time behavior and large time profile of viscous Hamilton-Jacobi equations in the periodic setting. Here, the diffusion matrix might be degenerate, which makes the problem more difficult and challenging. Based on joint works with Cagnetti, Gomes, Mitake.
May 27, 2021 -- Chi-Wang Shu -- Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs
In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently. When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete schemes paying attention only to spatial discretizations and leaving time $t$ continuous. It is then important to have a high order time discretization to main the stability properties of the semi-discrete schemes. In this talk we discuss several classes of high order time discretization, including the strong stability preserving (SSP) time discretization, which preserves strong stability from a stable spatial discretization with Euler forward, the implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly and the less stiff term (e.g. the convection term in such an equation) explicitly, for which strong stability can be proved under the condition that the time step is upper-bounded by a constant under suitable conditions, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performance of these schemes.
June 3, 2021 -- Anna Mazzucato -- Direct and inverse problems for a model of dislocations in geophysics
Abstract: I will discuss a model for dislocations in an elastic medium, modeling faults in the Earth's crust. The direct problem consists in solving a non-standard boundary value/interface problem for isotropic, in-homogeneous linear elasticity with piecewise Lipschitz Lame' parameters, for which we prove well-posedness and a double-layer potential representation for the solution if the coefficients jumps only along the fault. The non-linear inverse problem consists in determining the fault surface and slip vector from displacement measurements made at the surface. We prove uniqueness under some geometric conditions, using unique continuation results for systems.
We also establish shape derivative formulas under infinitesimal movements of the fault and changes in the slip. The application of the inverse problem is in fault monitoring and microseismicity. This is joint work with Andrea Aspri (Pavia University), Elena Beretta (Politechnico, Milan & NYU-Abu Dhabi), and Maarten de Hoop (Rice).
June 10, 2021 -- Philipp Grohs -- Deep Learning in Numerical Analysis
Abstract: The development of new classification and regression algorithms based on deep neural networks coined Deep Learning have had a dramatic impact in the areas of artificial intelligence, machine learning, and data analysis. More recently, these methods have been applied successfully to the numerical solution of partial differential equations (PDEs). However, a rigorous analysis of their potential and limitations is still largely open. In this talk we will survey recent results contributing to such an analysis. In particular I will present recent empirical and theoretical results supporting the capability of Deep Learning based methods to break the curse of dimensionality for several high dimensional PDEs, including nonlinear Black Scholes equations used in computational finance, Hamilton Jacobi Bellman equations used in optimal control, and stationary Schrödinger equations used in quantum chemistry. Despite these encouraging results, it is still largely unclear for which problem classes a Deep Learning based ansatz can be beneficial. To this end I will, in a second part, present recent work establishing fundamental limitations on the computational efficiency of Deep Learning based numerical algorithms that, in particular, confirm a previously empirically observed "theory-to-practice gap".
June 17, 2021 -- Apala Majumdar -- PDE problems in the Landau-de Gennes theory for Nematic Liquid Crystals
Abstract: Nematic liquid crystals are classical examples of partially ordered materials that combine fluidity with the order of crystalline solids. Nematics have long-range orientational order i.e. they are directional materials with special directions, referred to as directors. The Landau-de Gennes theory is one of the most celebrated and powerful continuum theories for nematic liquid crystals. In this talk, we review the mathematical framework for the Landau-de Gennes theory with emphasis on the Landau-de Gennes free energy and the associated Euler-Lagrange equations, which are typically a system of coupled, nonlinear partial differential equations. We review some recent results for boundary-value problems in the Landau-de Gennes theory, including results on the multiplicity, defect sets and asymptotic analysis of energy-minimizing solutions. We also describe the physical relevance of these solutions, followed by case studies of applications in the physical sciences and industry.
June 24, 2021 -- Yoichiro Mori -- Analysis of the Dynamics of Immersed Elastic Filaments in Stokes Flow
Abstract: We consider the problem of an elastic filament immersed in a 2D or 3D Stokes fluid. We first discuss the analysis of an immersed filament problem in a 2D Stokes fluid (the Peskin problem). We prove well-posedness and immediate regularization of the elastic filament configuration and discuss criteria for global existence. We will then discuss the immersed filament problem in a 3D Stokes fluid (the Slender Body problem). Here, it has not even been clear what the appropriate mathematical formulation of the problem should be. We propose a mathematical formulation for the Slender Body problem and discuss well-posedness for the stationary version of this problem. Furthermore, we prove that the Slender Body approximation, introduced by Keller and Rubinow in the 1980's and used widely in computation, provides an approximation to the Slender Body problem.
Past Webinar talks (2020 schedule)
Date | Speaker | Title (click to open videos/slides) | Related SN PDE Paper |
24/08 | Arnulf Jentzen (University of Münster) | Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep artificial neural networks | |
31/08 | Julio D. Rossi (Universidad de Buenos Aires) | A game theoretical approach for a nonlinear system driven by elliptic operators (video/slides) | |
07/09 | Thomas Bartsch (Universität Gießen) | Normalized solutions of nonlinear elliptic problems (video/slides) | https://rdcu.be/b6PnL |
14/09 | Weinan E (Princeton University) | PDE problems that arise from machine learning (video/slides) | |
21/09 | Claudio Muñoz (Universidad de Chile) | Understanding soliton dynamics in Boussinesq models (video/slides) | https://rdcu.be/b6oTf |
28/09 | Yihong Du (University of New England) | Long-time dynamics of the Fisher-KPP equation with nonlocal diffusion and free boundary (video/slides) | |
05/10 | George Em Karniadakis (Brown University and MIT) | From PINNs to DeepOnets: Approximating functions, functionals, and operators using deep neural networks | |
12/10 | Shi Jin (Shanghai Jiao Tong University) | Random Batch Methods for classical and quantum N-body problems (video/slides) | |
19/10 | Yoshikazu Giga (University of Tokyo) | On total variation flow type equations (video/slides) | https://rdcu.be/b8BtO |
26/10 | Susanna Terracini (Università di Torino) | Segregation, interaction of species and related free boundary problems | |
02/11 | Yanyan Li (Rutgers) | Gradient estimates for the insulated conductivity problem | |
09/11 | Jaeyoung Byeon (KAIST) | Nonlinear Schrödinger systems with large interaction forces between different components | |
16/11 | Edriss Titi (University of Cambridge) | The Inviscid Primitive Equations and the Effect of Rotation (video/slides) | |
23/11 | Pierre-Emmanuel Jabin (Penn State University) | Large stochastic systems of interacting particles (Video/Slides) | |
30/11 | Jose A. Carrillo (University of Oxford) | Nonlinear Aggregation-Diffusion Equations: Gradient Flows, Free Energies and Phase Transitions | |
07/12 | Marie E. Rognes (Simula) | The brain's waterscape (video/slides) | |
14/12 | Marie-Therese Wolfram (University of Warwick) | On mean-field models in pedestrian dynamics (video/slides) |
2020 Speakers and Abstracts
August 24, 2020 -- Arnulf Jentzen -- Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep artificial neural networks (slides)
Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE with a nonlinearity depending on the PDE solution can be solved approximatively without the curse of dimensionality.
August 31, 2020 -- Julio D. Rossi -- A game theoretical approach for a nonlinear system driven by elliptic operators (slides)
This talk is based on the interplay between partial differential equations and probability.
We find approximations using game theory to viscosity solutions to an elliptic system governed by two different operators (the Laplacian and the infinity Laplacian).
We analyze a game that combines Tug-of-War with Random Walks in two different boards with a positive probability of jumping from one board to the other and we prove that the value functions for this game converge uniformly to a viscosity solution of an elliptic system as the step size goes to zero.
In addition, we show uniqueness for the elliptic system using pure PDE techniques.
Joint work with A. Miranda (Buenos Aires).
September 7, 2020 -- Thomas Bartsch -- Normalized solutions of nonlinear elliptic problems (slides)
Abstract here
September 14, 2020 -- Weinan E -- PDE problems that arise from machine learning (slides)
Two kinds of PDE problems arise from machine learning. The continuous formulation of machine learning naturally gives rise to some very elegant and challenging PDE (more precisely partial differential and integral equations) problems. It is likely that understanding these PDE problems will become fundamental issues in the mathematical theory of machine learning.
Machine learning-based algorithms for PDEs also lead to new questions about these PDEs, for example, new kinds of a priori estimates that are suited for the machine learning model. I will discuss both kinds of problems.
September 21, 2020 -- Claudio Muñoz -- Understanding soliton dynamics in Boussinesq models (slides)
The purpose of this talk is to describe in simple terms the soliton problem for several Boussinesq models, including good, improved and abcd systems. The problem is not simple, because some particular unstable behavior present in each system above mentioned. The idea is to explain the particularities of each system, previous and recent results, and future research, in simple words.
September 28, 2020 -- Yihong Du -- Long-time dynamics of the Fisher-KPP equation with nonlocal diffusion and free boundary (slides)
We consider the Fisher-KPP equation with free boundary and "nonlocal diffusion". We show the problem is well-posed, and its long-time dynamical behavior is governed by a spreading-vanishing dichotomy. Moreover, we completely determine the spreading profile, which may have a finite spreading speed determined by a semi-wave problem, or have infinite spreading speed (accelerated spreading), according to whether a threshold condition on the kernel function is satisfied. Further more, for some typical kernel functions, we obtain sharp estimates of the spreading speed (whether finite or infinite).
October 5, 2020 -- George Em Karniadakis -- From PINNs to DeepOnets: Approximating functions, functionals, and operators using deep neural networks
We will present a new approach to develop a data-driven, learning-based framework for predicting outcomes of physical and biological systems, governed by PDEs, and for discovering hidden physics from noisy data. We will introduce a deep learning approach based on neural networks (NNs) and generative adversarial networks (GANs). We also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. Unlike other approaches that rely on big data, here we “learn” from small data by exploiting the information provided by the physical conservation laws, which are used to obtain informative priors or regularize the neural networks. We will also make connections between Gauss Process Regression and NNs and discuss the new powerful concept of meta-learning. We will demonstrate the power of PINNs for several inverse problems in fluid mechanics, solid mechanics and biomedicine including wake flows, shock tube problems, material characterization, brain aneurysms, etc, where traditional methods fail due to lack of boundary and initial conditions or material properties.
October 12, 2020 -- Shi Jin -- Random Batch Methods for classical and quantum N-body problems (slides)
We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions.
This method is also extended to quantum Monte-Carlo methods for N-body Schrodinger equation and will be shown to have significant gains in computational speed up over the classical Metropolis-Hastings algorithm and the Langevin dynamics based Euler-Maruyama method for statistical samplings of general distributions for interacting particles.
For quantum N-body Schrodinger equation, we also obtain, for pair-wise random interactions, a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is uniform in N > 1 and independent of the Planck constant \hbar. To this goal we need to introduce a new metric specially tailored to handle at the same time the difficulties pertaining to the small \hbar regime (classical limit), and those pertaining to the large N regime (mean-field limit).
This talk is based on joint works with Lei Li, Jian-Guo Liu, Francois Golse, Thierry Paul and Xiantao Li.
October 19, 2020 -- Yoshikazu Giga -- On total variation flow type equations (slides)
The classical total variation flow is the $L^2$ gradient flow of the total variation. The total variation of a function u is one-Dirichlet energy, i.e.,$ \int |Du| dx$. Different from the Dirichlet energy $\int |Du|^2 dx/2$, the energy density is singular at the place where the slope of the function u equals zero. Because of this structure, its gradient flow is actually non-local in the sense that the speed of slope zero part (called a facet) is not determined by infinitesimal quantity. Thus, the definition of a solution itself is a nontrivial issue even for the classical total variation flow. This becomes more serious if there is non-uniform driving force term.
Recently, there need to study various types of such equations. A list of examples includes the total variation map flow as well as the classical total variation flow and its fourth order version in image de-noising, crystalline mean curvature flow or fourth order total variation flow in crystal growth problems which are important models in materials science below roughening temperature.
In this talk, we survey recent progress on these equations with special emphasis on a crystalline mean curvature flow whose solvability was left open more than ten years. We shall give a global-in-time unique solvability in the level-set sense. It includes a recent extension when there is spatially non-uniform driving force term which is going to be published in the journal SN Partial Differential Equations. These last well-posedness results are based on my joint work with N. Požár (Kanazawa University) whose basic idea depends on my earlier joint work with M.-H. Giga (The University of Tokyo) and N. Požár.
October 26, 2020-- Susanna Terracini -- Segregation, interaction of species and related free boundary problems (slides)
Reaction-diffusion systems with strong interaction terms appear in many multi-species physical problems as well as in population dynamics, chemistry and material science. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example appears when looking for solitary wave solutions for Bose-Einstein condensates of two (or more) different hyperfine states which overlap in space. Typically the forces between particles in the same state are attractive while those between particles in different states can be either attractive or repulsive. If the condensates repel, they eventually separate spatially giving rise to a free boundary. This phenomenon is called phase separation and has been described in recent literature, both physical and mathematical.
One of the most interesting problems researchers investigate is when different phases of matter, populations, or clusters exist in a single space (i.e. in adjacent cells). Their interest focuses not only in how these different phases/populations/clusters interact with one another, but also on the properties of the boundaries separating them. The recent literature shows that the walls separating the different phases are geometrically tractable surfaces, as well as multiple junctions among them. This involves developing novel variational methods and geometric measure theory and free boundary tools. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of the nodal sets of solutions are of fundamental importance as well. We intend to focus on the most recent development of the theory in connection with problems featuring anomalous diffusions, long-range and non symmetric.
November 2, 2020 -- Yanyan Li -- Gradient estimates for the insulated conductivity problem (slides)
In this talk, we discuss the insulated conductivity problem with multiple inclusions embedded in a bounded domain in n-dimensional Euclidean space. The gradient of a solution may blow up as two inclusions approach each other. The optimal blow up rate was known in dimension n=2. It was not known whether the established upper bound of the blow up rates in higher dimensions were optimal.
We answer this question by improving the previously known upper bound of the blow up rates in dimension n>2.
This is a joint work with Zhuolun Yang.
November 9, 2020 -- Jaeyoung Byeon -- Nonlinear Schrödinger systems with large interaction forces between different components (slides)
There have been many studies on the asymptotic behavior of low energy solutions for a single elliptic equation as an involved parameter approaches to a threshold. In this case, the asymptotic behavior depends on a balance between the differential operator and nonlinearity, and their interaction with a geometry of a underlying domain. On the other hand, even though the elliptic systems coming from nonlinear Schrödinger systems have a simple looking interaction terms, even the construction of nontrivial low energy solutions is not easy in general since the Morse indices of the nontrivial solutions could be high depending types of interaction terms.
When the interaction forces between different components are very large, we believe that a relatively simpler structure we can see. Nevertheless, a wide variety of their asymptotic behavior we could imagine as various kinds of combination for the interaction between components might produce various effects on the asymptotic behavior. The general study for elliptic systems with large interaction forces is quite challenging.
In this talk, I would like to introduce my recent studies with collaborators on three components systems as basic steps to get general understanding for elliptic systems with large interaction forces.
November 16, 2020 -- Edriss Titi -- The Inviscid Primitive Equations and the Effect of Rotation (slides)
Large scale dynamics of the oceans and the atmosphere is governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous primitive equations are globally well-posed in Sobolev spaces. In this talk, I will discuss the ill-posedness in Sobolev spaces, the local well-posedness in the space of analytic functions, and the finite-time blowup of solutions to the three-dimensional inviscid PEs with rotation (Coriolis force). Eventually, I will also show, in the case of ``well-prepared" analytic initial data, the regularizing effect of the Coriolis force by providing a lower bound for the life-span 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 approximate the corresponding solution of the 3D inviscid PEs with the same initial data.
November 23, 2020 -- Pierre-Emmanuel Jabin -- Large stochastic systems of interacting particles
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.
November 30, 2020 -- Jose A. Carrillo -- Nonlinear Aggregation-Diffusion Equations: Gradient Flows, Free Energies and Phase Transitions
The main goal of this talk is to discuss the state-of-the-art in understanding the phenomena of phase transitions for a range of nonlinear Fokker-Planck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as Cucker-Smale 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.
December 7, 2020 -- Marie E. Rognes -- The brain's waterscape
Your brain has its own waterscape: whether you are reading or sleeping, fluid flows around or through the brain tissue and clears waste in the process. These physiological processes are crucial for the well-being of the brain. In spite of their importance we understand them but little. Mathematics and numerics could play a crucial role in gaining new insight. Indeed, medical doctors express an urgent need for modeling of water transport through the brain, to overcome limitations in traditional techniques. Surprisingly little attention has been paid to the numerics of the brain’s waterscape however, and fundamental knowledge is missing. In this talk, I will discuss mathematical models and numerical methods for the brain's waterscape across scales - from viewing the brain as a poroelastic medium at the macroscale and zooming in to studying electrical, chemical and mechanical interactions between brain cells at the microscale.
December 14, 2020 -- Marie-Therese Wolfram -- On mean-field models in pedestrian dynamics
In this talk I will start with a general overview on mean-field models for pedestrian dynamics, outlining the challenges in the derivation and the analysis of the corresponding PDE models. I will then illustrate how this continuum description can be used to understand the effect of inflow and outflow rates as well as the geometry on pedestrian density profiles. Finally I will present how the Bayesian framework can be used to identify parameters in mean field models and quantify uncertainty in those estimates using trajectory data.