Chair: Bob Pego, Carnegie Mellon University
Title: An obstacle problem for cell polarization
Featured Article: A parabolic free boundary problem arising in a model of cell polarization. SIAM J. Math. Anal. 53-1 (2021)
Featured Article Authors: A. Logioti, B. Niethammer, M. Röger and J. L. Velázquez
Abstract: We investigate a model for cell polarization under external stimulus where a diffusion equation in the inner cell is coupled to reaction diffusion equations on the cell membrane. In certain scaling limits we rigorously derive generalized obstacle type problems. For these limit systems we prove global stability of steady states and characterize the parameter regime for the onset of polarization. (Joint work with Anna Logioti (Bonn) Matthias Roeger (TU Dortmund) and Juan Velazquez (U Bonn))
Biography: Barbara Niethammer is Professor at the Institute for Applied Mathematics at the University of Bonn. Her research focuses on the analysis of problems with multiple scales and high-dimensional dynamical systems as well as on the study of long-time behaviour in models of mass aggregation. Before moving to Bonn she held faculty positions at the Humboldt-University of Berlin and the University of Oxford. She was invited sectional speaker at the ICM 2014, Emmy-Noether lecturer at the annual DMV meeting in 2019, plenary speaker at the SIAM annual meeting in 2020 and received the von-Mises-Prize of GAMM and the Whitehead prize of the LMS.
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Speaker: John Ball
Title: Some energy minimization problems for liquid crystals
Date of Presentation: June 4, 2020
Chair: Doug Arnold, University of Minnesota
Abstract: The talk will discuss some energy minimization problems for liquid crystals described at different levels of detail by the probability density function of molecular orientations, by a tensor average of this function (the de Gennes Q tensor theory), and by the expected orientation of molecules (the Oseen-Frank theory).
Biography: John Ball is Professor of Mathematics at Heriot-Watt University, Edinburgh, Emeritus Professor at the University of Oxford and Senior Fellow of the Hong Kong Institute for Advanced Study. His research interests lie in nonlinear analysis, the calculus of variations, infinite-dimensional dynamical systems, and their applications to materials science, including solid phase transformations and liquid crystals. He is a former President of the International Mathematical Union.
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Speaker: Benoit Perthame
Title: Multiphase models of living tissues and the Hele-Shaw limit
Date of Presentation: July 2, 2020
Chair: François Golse, Ecole Polytechnique
Abstract: The mechanical modeling of living tissues has attracted much attention in the last decade. Applications include tissue repair and growth models of solid tumors. These models contain several levels of complexity, both in terms of the biological and mechanical effects, and therefore in their mathematical description. Multiphase models describe the dynamics of several types of cells, liquid, fibers (extra-cellular matrix) and both compressible and incompressible models are used in the literature.
In this talk I shall discuss the analysis of multiphase models based on Darcy's assumption. The compactness issue leads us to use Aronson-Benilan estimate and to build new variants. I shall also discuss the incompressible limit in special cases and the associated free boundary problem.
Biography: Benoit Perthame is a professor of mathematics at Sorbonne Universite in Paris and former director of the Laboratoire Jacques-Louis Lions. Before, he has been a professor at Ecole Normale Superieure and the founder of the team Bang at Institut National de la Recherche en Informatique et Automatique, a team focussed on mathematical modeling in life sciences. His research activities concern partial differential equations, the mathematical objects which serve to relate variations in space and time as they arise in fluid flows and heat transfer. He has introduced a new striking relationship between dilute flows (Boltzman equation) and dense flows (Euler equations). Recently, he has shown the important role played by nonlinear PDEs in a number of problems from biology such as cell motion and cell colonies' self-organization, Darwinian evolution, modeling tumor growth and therapy, and neural networks. He was a plenary speaker ICIAM (Vancouver 2011) and at ICM 2014 (Seoul). He was elected to the French Academy of Sciences in 2017.
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Speaker: Felix Otto, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Title: The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows
Date of Presentation: Thursday, September 3, 2020 at 11:30 AM EDT
Chair: David Kinderlehrer, Carnegie Mellon University
Abstract: Flow of interfaces by mean curvature, in its multi-phase version, was first formulated in the context of grain growth in polycrystalline materials. The computationally efficient and very popular thresholding scheme for mean curvature flow by Osher et. al. can be naturally extended to such a multi-phase situation, even for surface tensions that depend on the lattice mismatch between the adjacent grains.
This extension relies on the gradient flow structure of mean curvature flow, and the interpretation of the thresholding scheme as a corresponding "minimizing movements'' scheme, that is, a sequence of variational problems naturally attached to the implicit time discretization of a gradient flow.
This interpretation also allows for a (conditional) convergence proof based on De Giorgi's ideas for gradient flows in metric spaces. The approach is similar to the convergence proof for the minimizing movement scheme by Almgren, Taylor and Wang, as given by Luckhaus et. al.
This is joint work with S. Esedoglu and T. Laux.
Biography: Felix Otto is director at the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany) since 2010. Before, he has been a professor at the University of California at Santa Barbara and at the Department of Applied Mathematics at the University of Bonn where he was the Managing director of the ‘Hausdorff Center for Mathematics’ from 2006–2009. His main expertise is in the applied analysis of partial differential equations and in the calculus of variations, lately also with randomness. He has worked on gradient flow structures of dissipative evolution equations, on pattern formation in ferromagnets, and on stochastic homogenization. He has received the Max Planck Research Prize, the Leibniz prize of the German Science Foundation, and the Collatz price of CICIAM. Presently, he is chair of the scientific committee at Oberwolfach Research Institute for Mathematics (MFO).
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Speaker Name: Laure Saint-Raymond
Speaker Affiliation: École Normale Supérieure de Lyon, France
Date and Time of Presentation: [RESCHEDULED] October 8th, 2020 11:30 am EDT
Title: Fluctuation theory in the Boltzmann-Grad limit
Chair: Walter Strauss, Brown University
Laure Saint-Raymond © French National Center for Scientific Research
Abstract: In this talk, I will discuss a long term project with T. Bodineau, I. Gallagher and S. Simonella on hard-sphere gases. In the low density limit, the empirical density obeys a law of large numbers and the dynamics is governed asymptotically by the (kinetic) Boltzmann equation. Deviations from this behavior are described by dynamical correlations, which we can fully characterize for short times. This provides both a fluctuating Boltzmann equation and large deviation asymptotics.
Biography: Laure Saint Raymond is currently a professor at the École Normale Supérieure de Lyon and a Fellow of the Institut Universitaire de France. Before, she has been a professor at the Université de Paris 6 – Pierre et Marie Curie. Her main contributions concern the transition between atomistic and continuous models for gas dynamics through rigorous mathematical analysis. Laure Saint-Raymond has obtained major results concerning the asymptotic theory of the Boltzmann equation in kinetic theory of gases. She has also studied problems of scale separation in the context of geophysical flows, especially for the wind-driven oceanic dynamics. She has been awarded many prizes, among which the SIAG/APDE Prize in 2006, joint with François Golse, the Prize of the European Mathematical Society in 2008, the Ruth Lyttle Satter Prize of the American Mathematical Society in 2009, the Fermat Prize in 2015 and the Bôcher Memorial Prize in 2020. She is a member of the FrenchAcademy of Sciences, the Academia Europae and the European Academy of Sciences.
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Speaker Name: Vlad Vicol
Speaker Affiliation: Courant Institute of Mathematical Sciences, NYU, USA
Date and Time of Presentation: November 5th, 2020 11:30 am EST
Chair: Edriss S. Titi, University of Cambridge, Texas A&M, and Wiezmann Institute
Title: Shock formation and vorticity creation for compressible Euler
Abstract: In this talk, I will discuss a long term project, joint with Tristan Buckmaster and Steve Shkoller, concerning 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. Via modulated self-similar variables, the blow-up time and location can be explicitly computed, and at the blow-up time, the solutions can be shown to have precisely Holder 1/3 regularity. Additionally, for the non-isentropic problem sounds waves interact with entropy waves to produce vorticity at the shock.
Biography: Vlad Vicol is currently a Professor of Mathematics at the Courant Institute for Mathematical Sciences, New York University. He received his Ph.D. in 2010 from the University of Southern California in 2010. He taught at the University of Chicago and Princeton University before joining the faculty at the Courant Institute in 2018. His research focuses on the analysis of partial differential equations arising in fluid dynamics, with an emphasis on problems motivated by hydrodynamic turbulence. Vicol is the recipient of an Alfred P. Sloan Research Fellowship, of the MCA Prize at the 2017 Mathematical Congress of the Americas, and a co-recipient of the 2019 Clay Research Award.
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Speaker Name: Alessio Figalli
Speaker Affiliation: ETH Zurich, Switzerland
Date and Time of Presentation: Thursday December 3, 2020 11:30 am EST
Chair: Camillo De Lellis (IAS Princeton, USA)
Title: Generic regularity in obstacle problems
Alessio Figalli
© ETH Zürich - Giulia Marthaler
Abstract: The classical obstacle problem consists of finding the equilibrium position of an elastic membrane whose boundary is held fixed and constrained to lie above a given obstacle. By classical results of Caffarelli, the free boundary is smooth 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 codimension 3 inside the free boundary, solving a conjecture of Schaeffer in dimension $n\leq 4$. This talk aims to give an overview of these results.
Biography: Alessio Figalli earned his doctorate in 2007 under the supervision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa and Cédric Villani at the École Normale Supérieure de Lyon.
He was a faculty at the University of Texas-Austin, before moving to ETH Zürich in 2016 as a chaired professor. Since 2019 he is the director of the “FIM-Institute for Mathematical Research” at ETH Zürich. Figalli has won an EMS Prize in 2012, the Peccot-Vimont Prize 2011, the Peccot 2012 of the Collège de France, the 2015 edition of the Stampacchia Medal, and the 2017 edition of the Feltrinelli Prize for mathematics, and has been appointed Nachdiplom Lecturer in 2014 at ETH Zürich. He was an invited speaker at the International Congress of Mathematicians 2014. In 2018 he won the Fields Medal for “his contributions to the theory of optimal transport, and its application to partial differential equations, metric geometry, and probability”.
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Speaker Name: Andrzej Swiech
Speaker Affiliation: Georgia Tech
Date and Time of Presentation: February 4, 2021, 11:30 am ET
Chair: Diogo Gomes, KAUST, Saudi Arabia
Title and Featured Article in SIAM Journal on Mathematical Analysis: Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures.
Featured Article Authors: W. Gangbo, S. Mayorga, A. Swiech
Abstract: We will discuss how certain Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures can be approximated by finite dimensional equations. The most interesting cases are convergence of viscosity solutions of HJB equations corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal control problems for systems of $n$ particles with a common noise, to the viscosity solution of a limiting HJB equation in the space of probability measures. The limiting HJB equation is interpreted in its ``lifted" form in a Hilbert space, which has a unique viscosity solution. When the Hamiltonian is convex in the gradient variable and equations are of first order, it can be proved that the viscosity solutions of the finite dimensional problems converge to the value function of a variational problem in $\mathcal{P}_2(\R^d)$ thus providing a representation formula for the solution of the limiting first order HJB equation. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces. The talk is based on a joint work with W. Gangbo and S. Mayorga.
Biography: Andrzej Swiech is a Professor of Mathematics at Georgia Institute of Technology in Atlanta (USA). He received his Ph.D. in 1993 from the University of California, Santa Barbara, under the direction of M.G. Crandall. His main interests are in fully nonlinear PDEs, PDEs in infinite-dimensional spaces, integro-PDEs, viscosity solutions, stochastic and deterministic optimal control, stochastic PDEs, differential games, mean-field games and control, calculus of variations, functional analysis. He received, jointly with S. Koike, 2010 JMSJ (Journal of The Mathematical Society of Japan) Outstanding Paper Prize.
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Speaker: Sergei Chernyshenko
Speaker Affiliation: Imperial College London, UK
Date and Time of Presentation: March 4, 2021 11:30 am EST
Chair: Ricardo Rosa, Universidade Federal do Rio de Janeiro, Brasil
Abstract: The problem of turbulence is the greatest unsolved problem of classical fluid dynamics. It is quite old.Similarly, the problem of optimizing over the cone of nonnegative polynomials was long believed to be too hard, until at the start of the millennium a breakthrough paved a way by linking it, through sum-of-squares polynomial optimisation, to semidefinite programming. And only a few years ago, auxiliary functionals were proposed with an ultimate goal of applying the fruits of this breakthrough to solving the problem of turbulence. The essence of these ideas is remarkably simple. The talk will cover the basics, and then a few questions will be reviewed deeper, including applications to systems of equations with energy-conserving quadratic nonlinearities, such as the Kuramoto–Sivashinsky and the Navier-Stokes equations.
Biography: Sergei Chernyshenko is a professor of aerodynamics at Imperial College London, UK. He had previous positions at the University of Southampton, UK, and at the Moscow State University, Russia. His area of expertise is theoretical fluid dynamics, with a scope ranging from rigorous mathematics to practical applications. He solved the long-standing problem of high-Re asymptotics of steady separated flow past a bluff body, and made contributions to the theory of organised structures in turbulent flows, methods of turbulent skin friction reduction, theory of scale interactionin near-wall turbulence, and theory of bounding time averages of dynamical systems. He received the USSR State Committee for People Education Award in research (shared), the Joukowski Award in aerodynamics (also shared), and was twice awarded the Russian State Stipend for Outstanding Scientists.
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Speaker: Sylvia Serfaty
Speaker Affiliation: Courant Institute of Mathematical Sciences, NYU
Date and Time of Presentation: April 1, 2021 11:30 am EDT
Chair: Didier Smets, Sorbonne Université, LJLL
Abstract: This talk will review older and more recent results on the analysis of vortices in the Ginzburg-Landau model of superconductivity, including description of energy minimizers and their vortex patterns in both two and three dimensions, and description of vortex dynamics.
Biography: Sylvia Serfaty is Silver Professor at the Courant Institute of Mathematical Sciences of New York University. Prior to this she has been Professor at the Université Pierre et Marie Curie (currently Sorbonne Université) at the Laboratoire Jacques-Louis Lions, and has held various appointments at the Courant Institute of NYU. She earned her BS and MS in Mathematics from the École Normale Supérieure in Paris, and her PhD from Université Paris Sud. She works in calculus of variations, nonlinear partial differential equations, and mathematical physics. She was a plenary speaker at the International Congress of Mathematicians in 2018, the recipient of the EMS and Henri Poincaré prizes, and is a member of the American Academy of Arts and Sciences.
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Speaker: Gigliola Staffilani
Speaker Affiliation: Massachusetts Institute of Technology
Date and Time of Presentation: May 6, 2021 11:30 am EDT
Chair: Luis Vega (BCAM, Spain)
Title: How much math do you need to know to “solve" an initial value problem?
Abstract: In this talk I will present some recent results concerning periodic solutions to nonlinear Schrodinger equations, and in doing so I will introduce a variety of mathematical techniques that range from harmonic and Fourier analysis to dynamical systems, from number theory to probability.
I will start with a derivation of this type of equation from a many body system, and I will discuss how Hamiltonian structures can be mapped through this derivation process. I will then move to the study of the long time dynamics of associated initial value problems, in particular I will concentrate on the notion of energy transfer. I will show how ideas from dynamical systems are fundamental to work through this analysis to obtain even relatively soft statements, and I will present some more recent results on the rigorous derivation of a wave kinetic equation for a certain multidimensional KdV type equation using a variety of tools such as Feynman diagrams, sharp dispersive estimates and improved combinatorial lemmata.
Biography: Gigliola Staffilani is the Abby Rockefeller Mauze Professor of Mathematics at MIT since 2007. She received the B.S. equivalent from the University of Bologna, and the M.S. and Ph.D. degrees from the University of Chicago. Following a Szegö Assistant Professorship at Stanford, she had faculty appointments at Stanford, Princeton and Brown before joining the MIT mathematics faculty in 2002. Professor Staffilani is an analyst, with a concentration on dispersive nonlinear PDEs. At Stanford, she received the Harold M. Bacon Memorial Teaching Award in 1997, and was given the Frederick E. Terman Award for young faculty in 1998. She was a Sloan fellow in 2000-02. Professor Staffilani was member of the Institute for Advanced Study in Princeton in 1996 and 2003, and member of the Radcliffe Institute for Advanced Study at Harvard University in 2010.
In 2013 Professor Staffilani was elected member of the Massachusetts Academy of Science and a fellow of the AMS, and in 2014 member of the American Academy of Arts and Sciences. In 2017 she received a Guggenheim fellowship and a Simons Fellowship in Mathematics. In 2018 she received the MIT Earll M. Murman Award for Excellence in Undergraduate Advising.
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Speaker: Yann Brenier
Speaker Affiliation: École Normale Supérieure, Paris
Date and Time of Presentation: June 3rd, 2021 11:30 am EDT
Abstract: We suggest a way of defining optimal transport of positive-semidefinite matrix-valued measures, inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main output of our work is a matricial analogue of the Hellinger-Kantorovich metric spaces.
Biography: Yann Brenier is Directeur de Recherches au CNRS, at Ecole Normale Superieure, Paris. Before, he has been a junior researcher at INRIA, a Hedrick assistant professor at UCLA, before becoming a professor jointly at Universite Paris 6 and Ecole Normale Superieure. Then, he became a Directeur de Recherches CNRS, first at Universite de Nice and later at Ecole Polytechnique, Palaiseau.
His research activities concern partial differential equations, with a special interest for Fluid Mechanics and Optimal Transport theory. He was a plenary speaker ICIAM (Sydney 2003), sectional speaker at ICM (Beijing 2002) and was awarded the Prize Petit d'Ormoy by the French Academy of Sciences in 2005.
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Speaker: Luis Silvestre
Speaker Affiliation: University of Chicago
Date and Time of Presentation: July 1st, 2021 11:30 a.m. EDT
Chair: Boyan Sirakov, Catholic University Rio de Janeiro - PUC-Rio
Title based on a Featured Article from SIAM Journal on Mathematical Analysis: Gaussian lower bounds for the Boltzmann equation without cut-off
Featured Article Authors: Cyril Imbert, Clément Mouhot, Luis Silvestre
Abstract: The Boltzmann equation models the evolution of densities of particles in a gas. Its global well posedness is a major open problem, facing comparable difficulties as similar questions for equations in fluids. With current techniques, we cannot rule out the possibility of a spontaneous emergence of a singularity in the form of infinite mass or energy density concentrating at some point in space. This work is part of a series of a priori estimates for the inhomogeneous non-cutoff Boltzmann equation that are conditional to bounds on macroscopic quantities. We establish a Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole assumption that hydrodynamic quantities (local mass, local energy and local entropy density) remain bounded. In the talk, we will discuss how this lower bound fits in the larger program of conditional estimates.
Biography: Luis Silvestre is a professor of mathematics at the University of Chicago. In 2005, he received his Ph.D. from the University of Texas at Austin. He was a Courant Instructor at NYU from 2005 to 2008. He got the Sloan research fellowship in 2009, a CAREER NSF grant in 2013, was an invited speaker at the ICM in 2014 and a plenary speaker in the Riviere-Fabes symposium in 2017. His interests include elliptic regularity, integro-differential equations, kinetic equations and conservation laws.
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Speaker: Roman Shvydkoy
Speaker Affiliation: University of Illinois, Chicago
Date and Time of Presentation: September 2nd, 2021 11:30 a.m. ET
Chair: Jose Antonio Carrillo (University of Oxford, UK)
Title: Analytical aspects of emergent dynamics in systems of collective behavior
Abstract: One of the challenging problems in the field of collective behavior is to understand global emergent phenomena arising from purely local communication between agents. In the context of alignment dynamics examples of such global phenomena include flock formations, reaching consensus of opinions, emergence of leaders, etc. For models based on purely local interactions the assumption of graph-, chain-, or hydrodynamic connectivity of the flock is often necessary, but not always guaranteed, to ensure collective outcome. In this talk we will give an overview of the subject and highlight two new approaches to alignment dynamics in disconnected or weakly connected flocks. One is based on a topological fractional diffusion that resembles the closest neighbor rule originating in various empirical studies, and another based on the use of random fluctuations and the hypocoercivity property for the resulting kinetic Fokker-Planck-Alignment equation.
Biography: Roman Shvydkoy is currently a Professor of Mathematics at the University of Illinois at Chicago. He held visiting positions at Courant Institute of Mathematical Sciences, Princeton University and Instituto de Ciencias Matematicas at Madrid. His research focuses on the analysis of PDEs arising in fluid dynamics and flocking. He is a recipient of the Simons Fellows in Mathematics award in 2018.
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Speaker: Monica Visan
Speaker Affiliation: University of California, Los Angeles
Date and Time of Presentation: October 7th, 2021 11:30 a.m. ET
Chair: Piero D'Ancona (University of Rome La Sapienza)
Abstract: We will survey a number of recent developments in the theory of completely integrable nonlinear dispersive PDE. These include a priori bounds, the orbital stability of multisolitons, well-posedness at optimal regularity, and the existence of dynamics for Gibbs distributed initial data. I will describe the basic objects that tie together these disparate results, as well as the diverse ideas required for each problem.
Biography: Monica Visan is a professor of mathematics at the University of California, Los Angeles. She received her undergraduate degree from the University of Bucharest in 2002. In 2006, she received her PhD degree from UCLA and was awarded a Clay Liftoff Fellowship.
She was a member at the Institute for Advanced Study in Princeton from 2006 to 2008, and an Assistant Professor at the University of Chicago from 2008 to 2009. In 2010, she was awarded a Harrington Fellowship and a Sloan Research Fellowship. She became a Kavli Fellow in 2010.
At UCLA, she received the Sorgenfrey Distinguished Teaching Award in 2018.
Professor Visan is an analyst working in nonlinear dispersive PDE, harmonic analysis, and completely integrable systems. Her early work centered around treating nonlinear dispersive equations at the scaling-critical regularity. More recently, she and her collaborators have developed new techniques that have proven effective at advancing the theory of completely integrable systems, including settling the optimal well-posedness problems for several much-studied equations such as the Korteweg-de Vries and the cubic nonlinear Schrodinger
equations.
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Speaker: Theodore Drivas
Speaker Affiliation: Stony Brook University
Date and Time of Presentation: November 4th, 2021 11:30 a.m. ET
Chair: Alexei Mailybaev (IMPA, Rio de Janeiro)
Abstract: We will discuss two types of one-dimensional compressible fluid equations; Navier-Stokes models with local dissipation in which the viscosity depends degenerately on the density and nonlocal models for collective dynamics which exhibit flocking behavior. For the local models, we prove large data global regularity for a class of equations covering viscous shallow water. Another result proves a conjecture of Peter Constantin on singularity formation for a model describing slender axisymmetric fluid jets. For the non-local models, we establish a continuation criterion which says that smooth solutions exist so long as no vacuum states form. The method of proof involves introducing a hierarchy of entropies to control the solution in terms of the minimum density. We also show that any weak solutions which obeying an entropy inequality exhibit flocking. This reports on joint work with P. Constantin, H. Nguyen, F. Pasqualotto and R. Shvydkoy.
Biography:
Theodore Drivas is currently an Assistant Professor of Mathematics at Stony Brook University. He received his B.S. degree from the University of Chicago in 2011 and his Ph.D. from Johns Hopkins University in 2017. Prior to arriving at Stony Brook in 2020, he was an NSF MSPRF postdoctoral fellow and subsequently Assistant professor at Princeton University. His research interests include mathematical fluid dynamics, turbulence theory, and dynamical systems.
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Speaker: Yao Yao
Speaker Affiliation: National University of Singapore
Date and Time of Presentation: December 2nd, 11:30 a.m. ET
Chair: Michele Coti Zelati (Imperial College London, UK)
Abstract: For some nonlocal PDEs, their steady states can be seen as critical points of some associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progress in the following equations, where the key is to carefully construct a suitable perturbation.
I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan). I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).
Biography: Yao Yao is currently an Associate Professor of Mathematics at the National University of Singapore. She received her BS degree from Peking University in 2007, and PhD degree in 2012 from UCLA. She was a Van Vleck Visiting Assistant Professor at University of Wisconsin-Madison in 2012-2015, and an Assistant Professor at Georgia Institute of Technology in 2015-2021. Her research focuses on the analysis of partial differential equations arising in mathematical biology and fluid dynamics, especially on the equations with a nonlocal transport term. She was a recipient of the NSF CAREER Award in 2018 and Sloan Research Fellowship in 2020.
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Speaker: François Golse
Speaker Affiliation: École polytechnique, France
Date and Time of Presentation: February 3rd, 2022, 11:30 a.m. EST
Chair: Mikaela Iacobelli, ETH Zurich, Switzerland
Abstract: In 1979, Dobrushin explained how Monge’s theory of optimal transport (1781) can be used to prove the mean-field limit for the classical dynamics of large particle systems. Is it possible to compare two quantum states, or a quantum state and a classical phase space density by some quantum analogue of optimal transport? in the affirmative, can one use such ideas to study the mean-field and the classical limits of quantum dynamics? This talk will review some recent results in this direction, based on joint works with E. Caglioti, C. Mouhot and T. Paul.
Biography: François Golse is a professor of mathematics at École polytechnique (Paris area, France). His research interests are PDEs and mathematical physics, especially kinetic models and their connection with fluid dynamics, and more recently the quantum dynamics of large particle systems in the classical and mean-field limits. With Laure Saint-Raymond, he has been awarded the first APDE prize of SIAM in 2006. He has given the 1993 Peccot Lectures at Collège de France (Paris) and the 2010 Harold Grad Lecture at the 27th Rarefied Gas Dynamics International Symposium (Pacific Grove, USA). He has been a plenary speaker at the 2004 European Congress of Mathematics (Stockholm) and an invited speaker at the 2006 International Congress of Mathematicians (Madrid).
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