Difference between revisions of "PDE Geometric Analysis seminar"
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'''March 28th, 2022.'''
'''March 28th, 2022.'''
[[Monica Visan]]; Format: online seminar via Zoom, Time:3:30PM-4:30
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30
Title: Determinants, commuting flows, and recent progress on
Title: Determinants, commuting flows, and recent progress on
Revision as of 20:28, 27 March 2022
The seminar's format will be a combination of online and in-person; we will make sure to update you as soon as we have more details available. First talk is tentatively scheduled for September 20th !
Some of the seminars will be held online. When that would be the case we would use the following zoom link
Meeting ID: 963 5468 1353 Passcode: 180680
PDE GA Seminar Schedule Fall 2021-Spring 2022
January 31th, 2022.
February 7th, 2022.
Jonah Duncan from John Hopkins University ; Format: online seminar via Zoom (see link above), Time: 3:30-4:30PM
Title: Estimates and regularity for the k-Yamabe equation
Abstract: The k-Yamabe problem is a fully nonlinear generalisation of the Yamabe problem, concerned with finding conformal metrics with constant k-curvature. In this talk, I will start by introducing the k-Yamabe problem, including a brief survey of established results and open problems. I will then discuss some recent work (joint with Luc Nguyen) on estimates and regularity for the k-Yamabe equation, addressing solutions in both the so-called positive and negative cones.
February 14th, 2022.
Sigurd Angenent; Format: online seminar via Zoom/ in person, Room:910, Time:3:30PM
Title: MCF after the Velázquez—Stolarski example.
Abstract: Velázquez (1995) constructed an example of a Mean Curvature Flow $M_t\subset\mathbb R^8$, $(-1<t<0)$ that blows up at the origin as $t\nearrow0$. Stolarski recently showed that in spite of the singularity the mean curvature on this solution is uniformly bounded. In joint work with Daskalopoulos and Sesum we constructed an extension of the Velázquez—Stolarski solution to positive times and show that it also has uniformly bounded mean curvature. In the talk I will describe the solutions and explain some of the ideas that show boundedness of the mean curvature.
February 21th, 2022.
Birgit Schoerkhuber; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
Time: 11:00 AM
Title: Nontrivial self-similar blowup in energy supercritical nonlinear wave equations
Abstract: Self-similar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finite-time blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new non-trivial self-similar solutions, which are completely explicit in all supercritical dimensions. Furthermore, we outline methods to analyse their stability. This involves a delicate spectral problem that we are able to solve rigorously in particular space dimensions. In these cases, we prove that the solutions are co-dimension one stable modulo translations. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).
February 28th, 2021.
Michael Hott; Format: online seminar via Zoom https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
Title: On quantum Boltzmann fluctuation dynamics at the presence of a BEC
Abstract: Boltzmann equations have served to describe transport properties in many instances. While usually heuristically justified by means of Markov processes, mathematically rigorous derivations from first principles started arising with the landmark works of Lanford and Cercignani, and there have been many important improvements ever since. To this day, it remains a very active mathematical and physical field. Starting with the Liouville-von Neumann equation for a weakly interacting highly condensed Bose gas in a finite periodic box, we will uncover a Boltzmann dynamics after identifying other dominating effects. Our work exhibits some parallels with a previous discussion by Zaremba-Niguni-Griffin. However, we will present an analytic dependence on physical parameters for the size of the individual terms in the expansion, for the size of errors, for the time of validity, as well as among physical parameters. We will also see how mathematical rigor uncovers important physical subtleties missed in the physical literature. Our work is the first rigorous derivation of a quantum Boltzmann equation from first principles. This is a joint work with Thomas Chen.
March 7th, 2022.
March 14th, 2022.
March 21th, 2022.
March 28th, 2022.
Monica Visan; Format: online seminar via Zoom, Time: 3:30PM-4:30
Meeting ID: 963 5468 1353 Passcode: 180680
Title: Determinants, commuting flows, and recent progress on completely integrable systems
Abstract: I 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 multisoliton solutions, 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.
April 4th, 2022.
Marcelo Disconzi; Format: online seminar via Zoom/ in person, Room:901, Time: 3:30PM
April 11th, 2022.
Dallas Albitron; Format: in person, Room:--, Time:--
Title: Non-uniqueness of Leray solutions of the forced Navier-Stokes equations
Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo
April 18th, 2022.
Loc Nguyen (UNCC); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.
April 25th, 2022.
TBA; Format: online seminar via Zoom/ in person, Room:--, Time:--
May 1th, 2022.
TBA; Format: online seminar via Zoom/ in person, Room:--, Time:--
September 20th, 2021.
Simion Schulz (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts
Abstract: We prove existence and Sobolev regularity of solutions of a nonlinear system of degenerate parabolic PDEs with self- and cross-diffusion, transport/confinement and nonlocal interaction terms. The macroscopic system of PDEs is formally derived from a large particle system and models the evolution of an arbitrary number of (e.g. biological) species with quadratic porous medium interactions in a bounded domain of arbitrary dimension. The cross-interactions are scaled by a coefficient on which a necessary smallness condition is imposed. The strategy of our proof relies on a fixed point argument, followed by a vanishing viscosity scheme. This is joint work with Maria Bruna (Cambridge), Luca Alasio (Paris VI), and Simone Fagioli (Università degli Studi dell'Aquila).
September 27th, 2021.
Dohyun Kwon (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM
Title: Volume-preserving crystalline and anisotropic mean curvature flow
Abstract: We consider the global existence of volume-preserving crystalline mean curvature flow in a non-convex setting. We show that a natural geometric property, associated with reflection symmetries of the Wulff shape, is preserved with the flow. Using this geometric property, we address the global existence and regularity of the flow for smooth anisotropies. For the non-smooth case, we establish global existence results for the types of anisotropies known to be globally well-posed. This is joint work with Inwon Kim (UCLA) and Norbert Požár (Kanazawa University).
October 4th, 2021.
Antoine Remind-Tiedrez (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries
Abstract: To describe the atmosphere on a synoptic scale (the scale at which high- and low-pressure systems are apparent on a weather map, for example) one may use the quasi-geostrophic equations, which are derived as a limit of the classical Boussinesq system under the assumptions of fast rotation and strong stratification. When incorporating the dynamics of water content in the atmosphere, a.k.a. moisture, one may then study the moist Boussinesq equations and its limit, the precipitating quasi-geostrophic equations.
These models are important for atmospheric scientists in light of the role that the water cycle plays in atmospheric dynamics, notably through energy budgeting (such as for example when atmospheric circulations are driven by laten heat release in storms). Mathematically, these models present interesting challenges due to the presence of boundaries, whose locations are a priori unknown, between phases saturated and unsaturated in water (schematically: boundaries between clouds and their surroundings).
In particular, while the (dry) quasi-geostrophic equations rely on the inversion of a Laplacian, this becomes a much trickier adversary in the presence of free boundaries. In this talk we will discuss how this nonlinear equation underpinning the precipitating quasi-geostrophic equations can be characterized using a variational formulation and we will describe the many benefits one may derive from this formulation.
October 11th, 2021.
October 18th, 2021.
Wojciech Ozanski (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM
Title: Well-posedness of logarithmic spiral vortex sheets.
Abstract: We will discuss a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We will discuss a recent result regarding a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we will explain that a spiral gives rise to such solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922, despite significant progress of the theory of vortex sheets and Birkhoff-Rott equations. We will also show well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane. This is joint work with P. Kokocki and T. Cieślak.
October 25th, 2021.
Maxwell Stolarski (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature
Abstract: Hypersurfaces moving by mean curvature flow often become singular in finite time. At this time, the flow may no longer be continued smoothly. The extension problem asks, "If M(t) is a solution to mean curvature flow defined up to time T, what conditions ensure that we may smoothly extend this solution to slightly later times?" For example, a result of Huisken says that if the 2nd fundamental forms of the evolving hypersurfaces remain uniformly bounded, then the mean curvature flow can be smoothly extended. One might then ask if a uniform bound on the mean curvature suffices to extend the flow. We'll discuss work that shows the answer is "no" in general, that is, there exist mean curvature flow solutions that become singular in finite time but have uniformly bounded mean curvature.
November 1th, 2021.
Lizhe Wan (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity
Abstract: This article is concerned with infinite depth gravity water waves with constant vorticity in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. We show that, for low-frequency solutions, the Benjamin-Ono equation gives a good and stable approximation to the system on the natural cubic time scale. The proof relies on refined cubic energy estimates and perturbative analysis.
November 8th, 2021.
Albert Ai (UW Madison);
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation
Abstract: In this talk we will consider the Cauchy problem for both low and high dispersive generalizations of the Benjamin-Ono equation. To address the nonlinear interactions, we use a pseudodifferential generalization of the gauge transform introduced by Tao for the original Benjamin-Ono equation. Further, we combine this with a paradifferential normal form. This approach allows for a much simpler functional setting, and improves the known low regularity well-posedness threshold across the range of the dispersive generalization. This is joint work with Grace Liu.
November 15th, 2021.
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP
Title: Global wellposedness for the energy-critical Zakharov system below the ground state
Abstract: The Zakharov system is a quadratically coupled system of a Schroedinger and a wave equation, which is related to the focussing cubic Schroedinger equation. We consider the associated Cauchy problem in the energy-critical dimension d=4 and prove that global well-posedness holds in the full (non-radial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr\"odinger equation with potentials solving the wave equation. A key ingredient in the non-radial setting is a bilinear Fourier extension estimate.
November 22th, 2021.
November 29th, 2021.
December 6th, 2021.
William Cooperman (University of Chicago); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM. Host: Hung Tran.
Title: Quantitative homogenization of Hamilton-Jacobi equations
Abstract: We are interested in the rate at which solutions to a Hamilton-Jacobi equation converge, in the large-scale limit, to the solution of the effective problem. We'll describe prior work in various settings where homogenization occurs (periodic or random in space, coercive or only "coercive on average" in momentum as in the G equation). We'll also use a theorem of Alexander, originally proved in the context of first-passage percolation, to improve the rate of convergence when an optimal control formulation is available (for example, in the G equation or when the Hamiltonian is convex and coercive).
December 13th, 2021.
PDE GA Seminar Schedule Fall 2020-Spring 2021
Welcome to the new mode of our PDEGA seminar this semester. Each week, we'll introduce to you two talks that are interesting and related to our interests. As the videos are already on Youtube or other platforms, you could choose to watch them whenever you want to; our goal here is merely to pick our favorite ones out of thousands of already available recorded talks.
Week 1 (9/1/2020-9/5/2020)
1. Paul Rabinowitz - The calculus of variations and phase transition problems. https://www.youtube.com/watch?v=vs3rd8RPosA
2. Frank Merle - On the implosion of a three dimensional compressible fluid. https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be
Week 2 (9/6/2020-9/12/2020)
1. Yoshikazu Giga - On large time behavior of growth by birth and spread. https://www.youtube.com/watch?v=4ndtUh38AU0
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
Week 3 (9/13/2020-9/19/2020)
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
Week 4 (9/20/2020-9/26/2020)
1. Robert M. Strain - Global mild solutions of the Landau and non-cutoff Boltzmann equation. https://www.youtube.com/watch?v=UWrCItk2euo&feature=youtu.be
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
Week 5 (9/27/2020-10/03/2020)
1. Isabelle Gallagher - From Newton to Boltzmann, fluctuations and large deviations. https://www.youtube.com/watch?v=BkrKkUVadDo
2. Connor Mooney - The Bernstein problem for elliptic functionals, https://www.youtube.com/watch?v=lSfnyfCL74c
Week 6 (10/04/2020-10/10/2020)
1. Felix Otto - The thresholding scheme for mean curvature flow and De Giorgi's ideas for gradient flows. https://www.youtube.com/watch?v=7FQsiZpQA7E
2. Inwon Kim - Evolution of star-shaped sets in Mean curvature flow with forcing http://www.birs.ca/events/2018/5-day-workshops/18w5033/videos/watch/201806190900-Kim.html
Week 7 (10/11/2020-10/17/2020)
1. Benoit Perthame - Multiphase models of living tissues and the Hele-Shaw limit. https://www.youtube.com/watch?v=UGVJnJCfw5s
2. Yifeng Yu - Properties of Effective Hamiltonians. https://www.youtube.com/watch?v=U06G4wjF-Hg
Week 8 (10/18/2020-10/24/2020)
1. Carlos Kenig - Asymptotic simplification for solutions of the energy critical nonlinear wave equation. https://youtu.be/jvzUqAxU8Xg
2. Kyeongsu Choi - Ancient mean curvature flows and singularity analysis. https://www.youtube.com/watch?v=Iu1iLjdFjKQ
Virtual Analysis and PDE Seminar (VAPS): https://sites.uci.edu/pdeonlineseminar/. First talk by Ovidiu Savin.
Week 9 (10/25/2020-10/31/2020)
1. John Ball - Some energy minimization problems for liquid crystals. https://www.youtube.com/watch?v=-j0jc-y7JzE
2. Tristan Buckmaster - Stable shock wave formation for the isentropic compressible Euler equations. https://stanford.zoom.us/rec/play/DwuT8rE-K1uJC0LghYPtsoaNmPBk9-P5EK4ZeWh1mVNJELRHn-ay-gOVXHSTRz_0X3iUZDBoUVYq8zfd.Tuqy8urKY4jESivm?continueMode=true&_x_zm_rtaid=GiRX307iT7encyYgIEgh9Q.1603308889393.b4a9b3af5c64cc9ca735cffbe25d8b7b&_x_zm_rhtaid=764
Week 10 (11/1/2020-11/7/2020)
1. Sylvia Serfaty - Mean-Field limits for Coulomb-type dynamics. https://www.youtube.com/watch?v=f7iSTnAe808&feature=youtu.be
2. Luc Nguyen - Symmetry and multiple existence of critical points in 2D Landau-de Gennes Q-tensor theory http://www.birs.ca/events/2017/5-day-workshops/17w5110/videos/watch/201705041518-Nguyen.html
Week 11 (11/8/2020-11/14/2020)
1. Andrzej Święch - Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures https://www.youtube.com/watch?v=KC514krtWAc
2. Alexandru Ionescu - On the nonlinear stability of shear flows and vortices, https://youtu.be/Zt_Izzi87V0
Week 12 (11/15/2020-11/21/2020)
1. Irene M. Gamba - Boltzmann type equations in a general framework: from the classical elastic flow, to gas mixtures, polyatomic gases, and more, https://youtu.be/fPlhAMGULtY
2. Andrej Zlatos - Euler Equations on General Planar Domains, https://www.youtube.com/watch?v=FdyyMZirRwk
Week 13 (11/22/2020-11/28/2020)
1. Camillo De Lellis - Flows of vector fields: classical and modern, https://www.youtube.com/watch?v=dVXSC3rtvok&feature=youtu.be
2. Wilfrid Gangbo - Analytical Aspect of Mean Field Games (Part 1/2), https://www.youtube.com/watch?v=KI5n6OYzzW8
Week 14 (11/29/2020-12/5/2020)
1. Juan Dávila - Leapfrogging vortex rings and other solutions with concentrated vorticity for the Euler equations, https://youtu.be/xfAKGc0IEUw
2. Yao Yao - Aggregation-diffusion equation: symmetry, uniqueness and non-uniqueness of steady states, https://www.youtube.com/watch?v=C_4qCimIMYc
Week 15 (12/6/2020-12/12/2020)
1. Pierre Gilles Lemarié-Rieusset - On weak solutions of the Navier-Stokes equations with infinite energy, https://www.youtube.com/watch?v=OeFJ6r-GLJc&feature=youtu.be
2. Albert Fathi - Weak KAM Theory: the connection between Aubry-Mather theory and viscosity solutions of the Hamilton-Jacobi equation, https://www.youtube.com/watch?v=0y8slhbQlTU
Week 1 (1/31/2021- 2/6/2021)
1. Emmanuel Grenier - instability of viscous shear layers https://www.youtube.com/watch?v=0_EG4VWIYvU&feature=youtu.be
2. Robert Pego - Dynamics and oscillations in models of coagulation and fragmentation https://www.youtube.com/watch?v=3712lImYP84
Week 2 ( 2/7/2021- 2/13/2021)
1. Ryan Hynd, The Hamilton-Jacobi equation, past and present https://www.youtube.com/watch?v=jR6paJf7aek
2. Jacob Bedrossian - Chaotic mixing of the Lagrangian flow map and the power spectrum of passive scalar turbulence in the Batchelor regime https://youtu.be/3lNQNsdlGTE
Colloquium (2/12/2021): Bobby Wilson (University of Washington). More information can be found here http://www.math.wisc.edu/wiki/index.php/Colloquia.
Week 3 ( 2/14/2021- 2/20/2021)
1. Diogo A. Gomes - Monotone MFGs - theory and numerics https://www.youtube.com/watch?v=lj1L7AHHY3s
2. Hao Jia - nonlinear asymptotic stability in two dimensional incompressible Euler equations https://youtu.be/KMf7K2sTLXg
Week 4 ( 2/21/2021- 2/27/2021)
1. Anne-Laure Dalibard - Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68
Week 5 ( 2/28/2021- 3/6/2021)
1. Inwon Kim - A variational scheme for Navier-Stokes Equations https://www.msri.org/workshops/944/schedules/29317
2. Robert L. Jerrard - Solutions of the Ginzburg–Landau equatons with vorticity concentrating near a nondegenerate geodesic https://www.youtube.com/watch?v=M0NQh2PET_k
Week 6 (3/7/2021-3/13/2021)
1. Ondřej Kreml - Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial datas https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231027-Kreml.html
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c
Week 7 (3/14/2021-3/20/2021)
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM
2. Alexis Vasseur - Instability of finite time blow-ups for incompressible Euler https://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011231000-Vasseur.html
Week 8 (3/21/2021- 3/27/2021)
1. Peter Sternberg - Variational Models for Phase Transitions in Liquid Crystals Based Upon Disparate Values of the Elastic Constants https://www.youtube.com/watch?v=4rSPsDvkTYs
2. François Golse - Half-space problem for the Boltzmann equation with phase transition at the boundary https://mysnu-my.sharepoint.com/:v:/g/personal/bear0117_seoul_ac_kr/ETGjasFQ7ylHu04qUz4KomYB98uMHLd-q96DOJGwbbEB0A
Week 9 (3/28/2021- 4/3/2021)
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM
2. Sergei Chernyshenko - Auxiliary functionals: a path to solving the problem of turbulence https://www.youtube.com/watch?v=NrF7n3MyCy4&list=PLf_ipOSbWC86n18q4JMn_1J04S90FpdeE&index=9
Week 10 (4/4/2021- 4/10/2021)
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8
Week 11(4/11/2021- 4/17/2021)
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo
2. David Gerard-Varet - On the effective viscosity of suspensions http://www.birs.ca/events/2020/5-day-workshops/20w5188/videos/watch/202011230644-Gerard-Varet.html
Week 12(4/18/2021- 4/24/2021)
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo
Week 13(4/25/2021- 5/1/2021)
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI
2. Stefania Patrizi - Chaotic Orbits for systems of nonlocal equations http://www.birs.ca/events/2017/5-day-workshops/17w5116/videos/watch/201704050939-Patrizi.html