Difference between revisions of "PDE Geometric Analysis seminar"

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The seminar will be held in room 901 of Van Vleck Hall on Mondays from 3:30pm - 4:30pm, unless indicated otherwise.
+
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
 +
 
 +
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]
 +
Passcode: 180680
 +
 
 +
 
 +
 
 +
[[Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.]]
 +
 
  
 
===[[Previous PDE/GA seminars]]===
 
===[[Previous PDE/GA seminars]]===
===[[Fall 2019-Spring 2020 | Tentative schedule for Fall 2019-Spring 2020]]===
 
  
== PDE GA Seminar Schedule Fall 2018-Spring 2019 ==
+
===[[Fall 2022-Spring 2023 | Schedule for Fall 2022-Spring 2023]]===
 +
 
 +
 
 +
 
 +
== PDE GA Seminar Schedule Fall 2022-Spring 2023 ==
 +
 
 +
=Fall 2022=   
 +
 
 +
 
 +
 
 +
'''September 12, 2022'''
 +
 
 +
[[No Seminar]]
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''September 20, 2022 (Tuesday)''' joint PDE and Analysis Seminar
 +
 
 +
[[Andrej Zlatos]] (UCSD). Host: Hung Tran.
 +
 
 +
Format: in-person. Time: 4-5PM, VV B139.
 +
 
 +
Title: Homogenization in front propagation models
 +
 
 +
Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit.  I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients.  These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.
 +
 
 +
 
 +
 
 +
 
 +
'''September 26, 2022 '''
 +
 
 +
[[Haotian Wu]] (The University of Sydney, Australia).  Host: Sigurd Angenent.
 +
 
 +
Format: in person, Time: 3:30pm-4:30pm VV 901 
 +
 
 +
<u>Title:</u>  ''Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up''
 +
 
 +
<u>Abstract:</u> The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).
 +
 
 +
 
 +
 
 +
 
 +
'''October 3, 2022'''
 +
 
 +
[[No Seminar]]
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
'''October 10, 2022'''
 +
 
 +
[[Alexander Kiselev]] (Duke). Host: Sergey Denisov.
 +
 
 +
Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).
 +
 
 +
Speaker: Sasha Kiselev (Duke)
 +
 
 +
Title: The flow of polynomial roots under differentiation
 +
Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work
 +
with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.
 +
 
 +
 
 +
 
 +
'''October 17, 2022'''
 +
 
 +
[[Nicolas Garca Trillos]] (Stats, UW Madison). Host: Hung Tran.
 +
 
 +
Format: in person, Time: 3:30pm-4:30pm VV 901 
 +
 
 +
'''Title:''' Analysis of adversarial robustness and of other problems in modern machine learning.
 +
 
 +
'''Abstract:''' Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures.
 +
 
 +
 
 +
 
 +
 
 +
'''October 24, 2022'''
 +
 
 +
[[TBA]]
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''October 31, 2022 ''' (Tentative date, which might be changed)
 +
 
 +
[[Yuan Gao]] (Purdue). Host: Hung Tran.
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
'''November 7, 2022 '''
 +
 
 +
[[TBA]]
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
'''November 14, 2022 '''
 +
 
 +
[[TBA]]
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
'''November 21, 2022 '''
 +
 
 +
[[Jason Murphy]] (Missouri S&T)
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
'''November 28, 2022 '''
 +
 
 +
[[No Seminar]]- Thanksgiving
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
'''December 5, 2022 '''
 +
 
 +
[[James Rowan]] (UC Berkeley)
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
'''December 12, 2022 '''
 +
 
 +
[[Calumn Rickard ]] UC Davis
 +
 
 +
Format:  , Time: 
 +
 
 +
Title:
 +
 
 +
Abstract:
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
 
 +
===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]===
 +
 
 +
== PDE GA Seminar Schedule Fall 2021-Spring 2022 ==
 +
 
 +
=Spring 2022=
 +
 
 +
 
 +
 
 +
 
 +
'''January  31th, 2022.'''
 +
 
 +
[[No Seminar]]
 +
 
 +
 
 +
'''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&mdash;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&mdash;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
 +
 
 +
Time:3:30PM-4:30PM
 +
 
 +
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.'''
 +
 
 +
[[ No Seminar]];
 +
 
 +
 
 +
'''March 14th, 2022.'''
 +
 
 +
[[Spring recess - No Seminar]];
 +
 
 +
 
 +
 
 +
 
 +
'''March 21th, 2022.'''
 +
 
 +
[[ No Seminar]];
 +
 
 +
 
 +
 
 +
'''March 28th, 2022.'''
 +
 
 +
[[Monica Visan]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM
 +
 
 +
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Meeting ID: 963 [[Tel:5468 1353|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
 +
 
 +
Title: General-relativistic viscous fluids.
 +
 
 +
 +
Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has
 +
 
 +
intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.
 +
 
 +
 
 +
 
 +
'''April 11th, 2022.'''
 +
 
 +
[[Dallas Albitron]]; Format: online seminar via Zoom, Time: 3:30PM-4:30PM
 +
 
 +
https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09
 +
 
 +
Meeting ID: 963 [[Tel:5468 1353|5468 1353]]
 +
Passcode: 180680
 +
 
 +
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.
 +
 
 +
Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.
 +
 
 +
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.
 +
 
 +
 
 +
 
 +
'''April 25th, 2022.'''
 +
 
 +
[[No seminar]]
 +
 
 +
 
 +
 
 +
'''May 2nd, 2022.'''
 +
 
 +
[[Alexei Gazca]]; Format: online seminar via Zoom, Time:3:30PM -4:30PM
 +
 
 +
Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness
  
 +
Abstract: In this talk, I will present some recent results obtained in
 +
collaboration with V. Patel (Oxford) in connection with a system
 +
describing a heat-conducting incompressible fluid. I will introduce the
 +
notion of a dissipative weak solution of the system and highlight the
 +
connections and differences to the existing approaches in the
 +
literature. One of the advantages of the proposed approach is that the
 +
solution satisfies a weak-strong uniqueness principle, which guarantees
 +
that the weak solution will coincide with the strong solution, as long
 +
as the latter exists; moreover, the solutions are constructed via a
 +
finite element approximation, leading (almost, not quite) to the first
 +
convergence result for the full system including viscous dissipation.
  
{| cellpadding="8"
 
!style="width:20%" align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!style="width:20%" align="left" | host(s)
 
  
|- 
 
|August 31 (FRIDAY),
 
| Julian Lopez-Gomez (Complutense University of Madrid)
 
|[[#Julian Lopez-Gomez | The theorem of characterization of the Strong Maximum Principle ]]
 
| Rabinowitz
 
  
|- 
+
=Fall 2021=
|September 10,
+
 
| Hiroyoshi Mitake (University of Tokyo)
+
'''September 20th, 2021.'''
|[[#Hiroyoshi Mitake | On approximation of time-fractional fully nonlinear equations ]]
+
 
| Tran
+
[[Simion Schulz]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
|-
+
 
|September 12 and September 14,
+
 
| Gunther Uhlmann (UWash)
+
Title: Existence and regularity for a system of porous medium equations with small cross-diffusion and nonlocal drifts
|[[#Gunther Uhlmann | TBA ]]
+
 
| Li
+
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 17,
+
 
| Changyou Wang (Purdue)
+
 
|[[#Changyou Wang | Some recent results on mathematical analysis of Ericksen-Leslie System ]]
+
'''September 27th, 2021.'''
| Tran
+
 
|-
+
[[Dohyun Kwon]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM
|Sep 28, Colloquium
+
 
| [https://www.math.cmu.edu/~gautam/sj/index.html Gautam Iyer] (CMU)
+
Title: Volume-preserving crystalline and anisotropic mean curvature flow
|[[#Sep 28: Gautam Iyer (CMU)| Stirring and Mixing ]]
+
 
| Thiffeault
+
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 1,
+
 
| Matthew Schrecker (UW)
+
'''October 4th, 2021.'''
|[[#Matthew Schrecker | Finite energy methods for the 1D isentropic Euler equations ]]
+
 
| Kim and Tran
+
[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
|-
+
 
|October 8,
+
Title: Variational formulation, well-posedness, and iterative methods for moist potential vorticity inversion: a nonlinear PDE from atmospheric dynamics with free boundaries
| Anna Mazzucato (PSU)
+
 
|[[#Anna Mazzucato | On the vanishing viscosity limit in incompressible flows ]]
+
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.
| Li and Kim
+
 
|- 
+
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).
|October 15,
+
 
| Lei Wu (Lehigh)
+
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.
|[[#Lei Wu | Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects ]]
+
 
| Kim
+
 
|-
+
'''October 11th, 2021.'''
|October 22,
+
 
| Annalaura Stingo (UCD)
+
[[No seminar]]
|[[#Annalaura Stingo | Global existence of small solutions to a model wave-Klein-Gordon system in 2D ]]
+
 
| Mihaela Ifrim
+
 
|- 
+
'''October 18th, 2021.'''
|October 29,
+
 
| Yeon-Eung Kim (UW)
+
[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM
|[[#Yeon-Eung Kim | Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties ]]
+
 
| Kim and Tran
+
Title: Well-posedness of logarithmic spiral vortex sheets.
|- 
+
 
|November 5,
+
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.
| Albert Ai (UC Berkeley)
+
 
|[[#Albert Ai | Low Regularity Solutions for Gravity Water Waves ]]
+
 
| Mihaela Ifrim
+
'''October 25th, 2021.'''
|-
+
 
|Nov 7 (Wednesday), Colloquium
+
[[Maxwell Stolarski]] (ASU); Format: in person seminar, Room: 901, Time: 3:30pm-4:30pm
| [http://math.mit.edu/~lspolaor/ Luca Spolaor] (MIT)
+
 
|[[#Nov 7: Luca Spolaor (MIT) |  (Log)-Epiperimetric Inequality and the Regularity of Variational Problems  ]]
+
Title: Mean Curvature Flow Singularities with Bounded Mean Curvature
| Feldman
+
 
|-
+
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.
|December 3, ''' Time: 3:00, Room: B223 Van Vleck '''
+
 
| Trevor Leslie (UW)
+
 
|[[#Trevor Leslie | Flocking Models with Singular Interaction Kernels ]]
+
'''November 1th, 2021.'''
| Kim and Tran
+
 
|-
+
[[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM
|December 10, ''' Time: 2:25, Room: B223 Van Vleck '''
+
 
|Serena Federico (MIT)
+
Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity
|[[#Serena Federico | Sufficient conditions for local solvability of some degenerate partial differential operators ]]
+
 
| Mihaela Ifrim
+
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.
|-
+
 
|December 10, Colloquium, '''Time: 4:00'''
+
 
| [https://math.mit.edu/~maxe/ Max Engelstein] (MIT)
+
 
|[[# Max Engelstein| The role of Energy in Regularity ]]
+
'''November 8th, 2021.'''
| Feldman
+
 
|-  
+
[[ Albert Ai]] (UW Madison);
|January 28,
+
 
| Ru-Yu Lai (Minnesota)
+
Title: Well-posedness for the dispersion-generalized Benjamin-Ono equation
|[[# Ru-Yu Lai | Inverse transport theory and related applications ]]
+
 
| Li and Kim
+
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.
|-
+
 
| Jan 31 '''4PM''',
+
 
| Dean Baskin (Texas A&M)
+
 
|[[# Dean Baskin | Radiation fields for wave equations]]
+
'''November 15th, 2021.'''
| Colloquium
+
 
|-
+
[[Sebastien Herr]] (Bielefeld University); Format: online seminar via Zoom, [[Time:10 AM]]
| February 4,
+
 
|
+
[[Please observe the time change! ]]
|[[# | No seminar (several relevant colloquiums in Jan/30-Feb/8)]]
+
 
|
+
Zoom Link: Register in advance for this meeting: https://uwmadison.zoom.us/meeting/register/tJcpcuqqqjMjE9VJ_-SaJ0gc6kS10CCTQTVP
|-
+
 
| February 11,
+
 
| Seokbae Yun (SKKU, long term visitor of UW-Madison)
+
Title: Global wellposedness for the energy-critical Zakharov system below the ground state
|[[# Seokbae Yun | The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation]]
+
 
| Kim
+
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.
|-
+
 
| February 18,  '''Room: VV B239'''
+
 
| Daniel Tataru (Berkeley)
+
 
|[[# Daniel Tataru | TBA ]]
+
'''November 22th, 2021.'''
| Ifrim
+
 
|-                                                                                                                                                          
+
[[No seminar]]
| February 19,
+
 
| Wenjia Jing (Tsinghua University)
+
 
|[[#Wenjia Jing | TBA ]]
+
'''November 29th, 2021.'''
| Tran
+
 
|-  
+
[[No seminar]]
|February 25,
+
 
| Xiaoqin Guo (UW)
+
 
|[[#Xiaoqin Guo | TBA ]]
+
'''December 6th, 2021.'''
| Kim and Tran
+
 
|-
+
[[William Cooperman]] (University of Chicago); Format: in-person seminar, Room:  901, Time: 3:30PM-4:30PM. Host: Hung Tran.
|March 4
+
 
| Vladimir Sverak (Minnesota)
+
Title: Quantitative homogenization of Hamilton-Jacobi equations
|[[#Vladimir Sverak | TBA(Wasow lecture) ]]
+
 
| Kim
+
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).
|-  
+
 
|March 11
+
 
| Jonathan Luk (Stanford)
+
'''December  13th, 2021.'''
|[[#Jonathan Luk | TBA  ]]
+
 
| Kim
+
 
|-
+
[[No seminar ]]
|March 12, '''4:00 p.m. in VV B139'''
+
 
| Trevor Leslie (UW-Madison)
+
 
|[[# Trevor Leslie| TBA ]]
+
 
| Analysis seminar
+
== 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. 
|March 18,
+
 
| Spring recess (Mar 16-24, 2019)
+
'''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
|March 25 (open)
+
 
| Open 
+
2. Frank Merle - On the implosion of a three dimensional compressible fluid.
|[[# Open  |Open  ]]
+
https://www.youtube.com/watch?v=5wSNBN0IRdA&feature=youtu.be 
+
 
|-  
+
'''Week 2 (9/6/2020-9/12/2020)'''
|April 1
+
 
| Zaher Hani (Michigan)
+
1. Yoshikazu Giga - On large time behavior of growth by birth and spread.
|[[#Zaher Hani | TBA  ]]
+
https://www.youtube.com/watch?v=4ndtUh38AU0
| Ifrim
+
 
|-  
+
2. Tarek Elgindi - Singularity formation in incompressible fluids. https://youtu.be/29zUjm7xFlI
|April 8 (open)
+
 
| Open 
+
 
|[[#Open | Open ]]
+
 
+
'''Week 3 (9/13/2020-9/19/2020)'''
|-
+
 
|April 15,
+
1. Eugenia Malinnikova - Two questions of Landis and their applications. https://www.youtube.com/watch?v=lpTsW1noWTQ
| Yao Yao (Gatech)
+
 
|[[#Yao Yao | TBA ]]
+
2. Pierre Germain - On the derivation of the kinetic wave equation. https://youtu.be/ZbCjKwQ3KcE
| Tran
+
 
|-  
+
 
|April 22,
+
 
| Jessica Lin (McGill University)
+
'''Week 4 (9/20/2020-9/26/2020)'''
|[[#Jessica Lin | TBA ]]
+
 
| Tran
+
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
|-  
+
 
|April 29,
+
2. Elena Kosygina - Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension https://www.youtube.com/watch?v=tVZv0ftT3PM
| Beomjun Choi (Columbia)
+
 
|[[#Beomjun Choi  | Evolution of non-compact hypersurfaces by inverse mean curvature]]
+
 
| Angenent
+
 
|}
+
'''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
 +
 
 +
 
 +
'''Spring 2021'''
 +
 
 +
'''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
  
== Abstracts ==
 
  
===Julian Lopez-Gomez===
 
  
Title: The theorem of characterization of the Strong Maximum Principle
+
'''Week 4 ( 2/21/2021- 2/27/2021)'''
  
Abstract: The main goal of this talk is to discuss the classical (well known) versions of the strong maximum principle of Hopf and Oleinik, as well as the generalized maximum principle of Protter and Weinberger. These results serve as steps towards the theorem of characterization of the strong maximum principle of the speaker, Molina-Meyer and Amann, which substantially generalizes a popular result of Berestycki, Nirenberg and Varadhan.
+
1. Anne-Laure Dalibard Boundary layer methods in semilinear fluid equations https://www.msri.org/workshops/944/schedules/29309
  
===Hiroyoshi Mitake===
+
2. Gui-Qiang G. Chen - On Nonlinear PDEs of Mixed Elliptic-Hyperbolic Type: Analysis and Connections https://www.youtube.com/watch?v=W3sa-8qtw68
Title: On approximation of time-fractional fully nonlinear equations
 
  
Abstract: Fractional calculus has been studied extensively these years in wide fields. In this talk, we consider time-fractional fully nonlinear equations. Giga-Namba (2017) recently has established the well-posedness (i.e., existence/uniqueness) of viscosity solutions to this equation. We introduce a natural approximation in terms of elliptic theory and prove the convergence. The talk is based on the joint work with Y. Giga (Univ. of Tokyo) and Q. Liu (Fukuoka Univ.)  
+
'''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
  
===Changyou Wang===
+
'''Week 6 (3/7/2021-3/13/2021)'''
  
Title: Some recent results on mathematical analysis of Ericksen-Leslie System
+
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
  
Abstract: The Ericksen-Leslie system is the governing equation  that describes the hydrodynamic evolution of nematic liquid crystal materials, first introduced by J. Ericksen and F. Leslie back in 1960's. It is a coupling system between the underlying fluid velocity field and the macroscopic average orientation field of the nematic liquid crystal molecules. Mathematically, this system couples the Navier-Stokes equation and the harmonic heat flow into the unit sphere. It is very challenging to analyze such a system by establishing the existence, uniqueness, and (partial) regularity of global (weak/large) solutions, with many basic questions to be further exploited. In this talk, I will report some results we obtained from the last few years.
+
2. Rita Ferreira - Homogenization of a stationary mean-field game via two-scale convergence https://www.youtube.com/watch?v=EICMVmt5o9c
  
===Matthew Schrecker===
 
  
Title: Finite energy methods for the 1D isentropic Euler equations
+
'''Week 7 (3/14/2021-3/20/2021)'''
  
Abstract: In this talk, I will present some recent results concerning the 1D isentropic Euler equations using the theory of compensated compactness in the framework of finite energy solutions. In particular, I will discuss the convergence of the vanishing viscosity limit of the compressible Navier-Stokes equations to the Euler equations in one space dimension. I will also discuss how the techniques developed for this problem can be applied to the existence theory for the spherically symmetric Euler equations and the transonic nozzle problem. One feature of these three problems is the lack of a priori estimates in the space $L^\infty$, which prevent the application of the standard theory for the 1D Euler equations.
+
1. Sergey Denisov - Small scale formation in 2D Euler dynamics https://www.youtube.com/watch?v=7ffUgTC34tM
  
===Anna Mazzucato===
+
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
  
Title: On the vanishing viscosity limit in incompressible flows
 
  
Abstract: I will discuss recent results on the  analysis of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations converge to solutions of the Euler equations, for incompressible fluids when walls are present. At small viscosity, a viscous boundary layer arise near the walls where large gradients of velocity and vorticity  may form and propagate in the bulk (if the boundary layer separates). A rigorous justification of Prandtl approximation, in absence of analyticity or monotonicity of the data, is available essentially only in the linear or weakly linear regime under no-slip boundary conditions.  I will present in particular a detailed analysis of the boundary layer for an Oseen-type equation (linearization around a steady Euler flow) in general smooth domains.
 
  
===Lei Wu===
+
'''Week 8 (3/21/2021- 3/27/2021)'''
  
Title: Hydrodynamic Limits in Kinetic Equations with Boundary Layer Effects
+
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
  
Abstract: Hydrodynamic limits concern the rigorous derivation of fluid equations from kinetic theory. In bounded domains, kinetic boundary corrections (i.e. boundary layers) play a crucial role. In this talk, I will discuss a fresh formulation to characterize the boundary layer with geometric correction, and in particular, its applications in 2D smooth convex domains with in-flow or diffusive boundary conditions. We will focus on some newly developed techniques to justify the asymptotic expansion, e.g. weighted regularity in Milne problems and boundary layer decomposition.
+
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
  
  
===Annalaura Stingo===
 
  
Title: Global existence of small solutions to a model wave-Klein-Gordon system in 2D
+
'''Week 9 (3/28/2021- 4/3/2021)'''
  
Abstract: This talk deals with the problem of global existence of solutions to a quadratic coupled wave-Klein-Gordon system in space dimension 2, when initial data are small, smooth and mildly decaying at infinity.Some physical models, especially related to general relativity, have shown the importance of studying such systems. At present, most of the existing results concern the 3-dimensional case or that of compactly supported initial data. We content ourselves here with studying the case of a model quadratic quasi-linear non-linearity, that expresses in terms of « null forms »  .
+
1. Susan Friedlander - Kolmogorov, Onsager and a stochastic model for turbulence https://www.youtube.com/watch?v=xk3KZQ-anDM
Our aim is to obtain some energy estimates on the solution when some Klainerman vector fields are acting on it, and sharp uniform estimates. The former ones are recovered making systematically use of normal forms’ arguments for quasi-linear equations, in their para-differential version, whereas we derive the latter ones by deducing a system of ordinary differential equations from the starting partial differential system. We hope this strategy will lead us in the future to treat the case of the most general non-linearities.
 
  
===Yeon-Eung Kim===
+
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
  
Title: Construction of solutions to a Hamilton-Jacobi equation with a maximum constraint and some uniqueness properties
 
  
A biological evolution model involving trait as space variable has a interesting feature phenomena called Dirac concentration of density as diffusion coefficient vanishes. The limiting equation from the model can be formulated by Hamilton Jacobi equation with a maximum constraint. In this talk, I will present a way of constructing a solution to a constraint Hamilton Jacobi equation together with some uniqueness and non-uniqueness properties.
+
'''Week 10 (4/4/2021- 4/10/2021)'''
  
===Albert Ai===
+
1. Camillo De Lellis - Transport equations and ODEs with nonsmooth coefficients https://www.msri.org/workshops/945/schedules/29235
  
Title: Low Regularity Solutions for Gravity Water Waves
+
2. Weinan E - PDE problems that arise from machine learning https://www.youtube.com/watch?v=5rb8DJkxfg8
  
Abstract: We consider the local well-posedness of the Cauchy problem for the gravity water waves equations, which model the free interface between a fluid and air in the presence of gravity. It has been known that by using dispersive effects, one can lower the regularity threshold for well-posedness below that which is attainable by energy estimates alone. Using a paradifferential reduction of Alazard-Burq-Zuily and low regularity Strichartz estimates, we apply this idea to the well-posedness of the gravity water waves equations in arbitrary space dimension. Further, in two space dimensions, we discuss how one can apply local smoothing effects to further extend this result.
+
'''Week 11(4/11/2021- 4/17/2021)'''
  
===Trevor Leslie===
+
1. Marian Gidea - Topological methods and Hamiltonian instability https://youtu.be/aMN7zJZavDo
  
Title: Flocking Models with Singular Interaction Kernels
+
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
  
Abstract: Many biological systems exhibit the property of self-organization, the defining feature of which is coherent, large-scale motion arising from underlying short-range interactions between the agents that make up the system.  In this talk, we give an overview of some simple models that have been used to describe the so-called flocking phenomenon.  Within the family of models that we consider (of which the Cucker-Smale model is the canonical example), writing down the relevant set of equations amounts to choosing a kernel that governs the interaction between agents.  We focus on the recent line of research that treats the case where the interaction kernel is singular.  In particular, we discuss some new results on the wellposedness and long-time dynamics of the Euler Alignment model and the Shvydkoy-Tadmor model.
+
'''Week 12(4/18/2021- 4/24/2021)'''
  
===Serena Federico===
+
1. Takis Souganidis - Phase-field models for motion by mean curvature - 1 https://www.youtube.com/watch?v=fH8ygVAZm-0
  
Title: Sufficient conditions for local solvability of some degenerate partial differential operators
+
2. Nader Masmoudi - Inviscid Limit and Prandtl System, https://youtu.be/tLg3HwVDlOo
  
Abstract: In  this  talk  we  will  give  sufficient  conditions  for  the  local  solvability  of  a  class  of degenerate second order linear partial differential operators with smooth coefficients. The class under consideration, inspired by some generalizations of the Kannai operator, is characterized by the presence of a complex subprincipal symbol.  By giving suitable conditions on the subprincipal part and using the technique of a priori estimates,  we will show that the operators in the class are at least $L^2$ to $L^2$ locally solvable.
+
'''Week 13(4/25/2021- 5/1/2021)'''
  
===Max Engelstein===
+
1. James Stone - Astrophysical fluid dynamics https://youtu.be/SlPSa37QMeI
  
Title: The role of Energy in Regularity
+
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
  
Abstract: The calculus of variations asks us to minimize some energy and then describe the shape/properties of the minimizers. It is perhaps a surprising fact that minimizers to ``nice" energies are more regular than one, a priori, assumes. A useful tool for understanding this phenomenon is the Euler-Lagrange equation, which is a partial differential equation satisfied by the critical points of the energy.
 
  
However, as we teach our calculus students, not every critical point is a minimizer. In this talk we will discuss some techniques to distinguish the behavior of general critical points from that of minimizers. We will then outline how these techniques may be used to solve some central open problems in the field.
 
  
We will then turn the tables, and examine PDEs which look like they should be an Euler-Lagrange equation but for which there is no underlying energy. For some of these PDEs the solutions will regularize (as if there were an underlying energy) for others, pathological behavior can occur.
+
{| cellpadding="8"
 +
!style="width:20%" align="left" | date 
 +
!align="left" | speaker
 +
!align="left" | title
 +
!style="width:20%" align="left" | host(s)
 +
|- 
 +
|}
  
===Seokbae Yun===
+
== Abstracts ==
Title: The propagations of uniform upper bounds fo the spatially homogeneous relativistic Boltzmann equation
 
  
Abstract: In this talk, we consider the propagation of the uniform upper bounds
+
===  ===
for the spatially homogenous relativistic Boltzmann equation. For this, we establish two
 
types of estimates for the the gain part of the collision operator: namely, a potential
 
type estimate and a relativistic hyper-surface integral estimate. We then combine them
 
using the relativistic counter-part of the Carlemann representation to derive a uniform
 
control of the gain part, which gives the desired propagation of the uniform bounds of
 
the solution. Some applications of the results are also considered. This is a joint work
 
with Jin Woo Jang and Robert M. Strain.
 
  
===Beomjun Choi===
+
Title:  
In this talk, we first introduce the inverse mean curvature flow and its well known application in the the proof of Riemannian Penrose inequality by Huisken and Ilmanen. Then our main result on the existence and behavior of convex non-compact solution will be discussed.  
 
  
The key ingredient is a priori interior in time estimate on the inverse mean curvature in terms of the aperture of supporting cone at infinity. This is a joint work with P. Daskalopoulos and I will also mention the recent work with P.-K. Hung concerning the evolution of singular hypersurfaces.
+
Abstract:

Latest revision as of 10:52, 4 October 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

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|5468 1353] Passcode: 180680


Please make sure to check the seminar webpage regularly so you will be constantly correctly informed on the format and time of the seminar.


Previous PDE/GA seminars

Schedule for Fall 2022-Spring 2023

PDE GA Seminar Schedule Fall 2022-Spring 2023

Fall 2022

September 12, 2022

No Seminar



September 20, 2022 (Tuesday) joint PDE and Analysis Seminar

Andrej Zlatos (UCSD). Host: Hung Tran.

Format: in-person. Time: 4-5PM, VV B139.

Title: Homogenization in front propagation models

Abstract: Homogenization is a general principle that the dynamics of physical processes occurring in periodic or random environments often become effectively homogeneous in the long-time-large-scale limit.  I will presents results showing that homogenization occurs for reaction-diffusion equations with both time-periodic-spatially-random and space-time-random KPP reactions and coefficients.  These results rely on two crucial new tools: virtual linearity of KPP reaction-diffusion dynamics and a non-autonomous versions of Kingman’s subadditive ergodic theorem.



September 26, 2022

Haotian Wu (The University of Sydney, Australia). Host: Sigurd Angenent.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

Abstract: The mean curvature flow (MCF) deforms a hypersurface in the direction of its mean curvature vectors. Singularities in mean curvature flow can form in either finite or infinite time. We present some results concerning the precise asymptotics of non-compact MCF solutions with either Type-IIa (in finite time) or Type-IIb (in infinite time) curvature blow-up. This is based on joint works with Jim Isenberg (Oregon) and Zhou Zhang (Sydney).



October 3, 2022

No Seminar

Format: , Time:

Title:

Abstract:



October 10, 2022

Alexander Kiselev (Duke). Host: Sergey Denisov.

Date/time/place: Monday, October 10, 3:30-4:30 pm, VV 901 (if more space will be needed, we have VV B119 to migrate to).

Speaker: Sasha Kiselev (Duke)

Title: The flow of polynomial roots under differentiation Abstract: The question of how polynomial roots move under differentiation is classical. Contributions to this subject have been made by Gauss, Lucas, Marcel Riesz, Polya and many others. In 2018, Stefan Steinerberger derived formally a PDE that should describe the dynamics of roots under differentiation in certain situations. The PDE in question is of hydrodynamic type and bears a striking resemblance to the models used in mathematical biology to describe collective behavior and flocking of various species- such as fish, birds or ants. I will discuss joint work with Changhui Tan in which we establish global regularity of Steinerberger's equation and make a rigorous connection between its solutions and evolution of roots under differentiation for a class of trigonometric polynomials.


October 17, 2022

Nicolas Garca Trillos (Stats, UW Madison). Host: Hung Tran.

Format: in person, Time: 3:30pm-4:30pm VV 901

Title: Analysis of adversarial robustness and of other problems in modern machine learning.

Abstract: Modern machine learning methods, in particular deep learning approaches, have enjoyed unparalleled success in a variety of challenging application fields like image recognition, medical image reconstruction, and natural language processing. While a vast majority of previous research in machine learning mainly focused on constructing and understanding models with high predictive power, consensus has emerged that other properties like stability and robustness of models are of equal importance and in many applications essential. This has motivated researchers to investigate the problem of adversarial training (or how to make models robust to adversarial attacks), but despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about it that remain relatively unexplored. In this talk, I will take an analytical perspective on the adversarial robustness problem and explore three questions: 1)What is the connection between adversarial robustness and inverse problems?, 2) Can we use analytical tools to find lower bounds for adversarial robustness problems?, 3) How do we use modern tools from analysis and geometry to solve adversarial robustness problems? At its heart, this talk is an invitation to view adversarial machine learning through the lens of mathematical analysis, showcasing a variety of rich connections with perimeter minimization problems, optimal transport, mean field PDEs of interacting particle systems, and min-max games in spaces of measures.



October 24, 2022

TBA

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Title:

Abstract:



October 31, 2022 (Tentative date, which might be changed)

Yuan Gao (Purdue). Host: Hung Tran.

Format: , Time:

Title:

Abstract:


November 7, 2022

TBA

Format: , Time:

Title:

Abstract:


November 14, 2022

TBA

Format: , Time:

Title:

Abstract:



November 21, 2022

Jason Murphy (Missouri S&T)

Format: , Time:

Title:

Abstract:


November 28, 2022

No Seminar- Thanksgiving




December 5, 2022

James Rowan (UC Berkeley)

Format: , Time:

Title:

Abstract:



December 12, 2022

Calumn Rickard UC Davis

Format: , Time:

Title:

Abstract:





Schedule for Fall 2021-Spring 2022

PDE GA Seminar Schedule Fall 2021-Spring 2022

Spring 2022

January 31th, 2022.

No Seminar


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

Time:3:30PM-4:30PM

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.

No Seminar;


March 14th, 2022.

Spring recess - No Seminar;



March 21th, 2022.

No Seminar;


March 28th, 2022.

Monica Visan; Format: online seminar via Zoom, Time: 3:30PM-4:30PM

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|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

Title: General-relativistic viscous fluids.


Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has

intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.


April 11th, 2022.

Dallas Albitron; Format: online seminar via Zoom, Time: 3:30PM-4:30PM

https://uwmadison.zoom.us/j/96354681353?pwd=SGlwUW1ockp6YklYYlppbDFZcW8zdz09

Meeting ID: 963 [1353|5468 1353] Passcode: 180680

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.

Title: The Carleman-based convexification approach for the 3D inverse scattering problem with experimental data.

Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.


April 25th, 2022.

No seminar


May 2nd, 2022.

Alexei Gazca; Format: online seminar via Zoom, Time:3:30PM -4:30PM

Title: Heat-conducting Incompressible Fluids and Weak-Strong Uniqueness

Abstract: In this talk, I will present some recent results obtained in collaboration with V. Patel (Oxford) in connection with a system describing a heat-conducting incompressible fluid. I will introduce the notion of a dissipative weak solution of the system and highlight the connections and differences to the existing approaches in the literature. One of the advantages of the proposed approach is that the solution satisfies a weak-strong uniqueness principle, which guarantees that the weak solution will coincide with the strong solution, as long as the latter exists; moreover, the solutions are constructed via a finite element approximation, leading (almost, not quite) to the first convergence result for the full system including viscous dissipation.


Fall 2021

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.

No seminar


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.

Sebastien Herr (Bielefeld University); Format: online seminar via Zoom, Time:10 AM

Please observe the time change!

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.

No seminar


November 29th, 2021.

No seminar


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.


No seminar


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


Spring 2021

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


date speaker title host(s)

Abstracts

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