PDE Geometric Analysis seminar: Difference between revisions
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Format: In person, Time: 3:30-4:30PM. | Format: In person, Time: 3:30-4:30PM. | ||
'''Title: | '''Title: ''' Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion | ||
'''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor. | '''Abstract:''' We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor. | ||
Revision as of 05:32, 21 April 2023
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
Previous PDE/GA seminars
Schedule for Fall 2023-Spring 2024
PDE GA Seminar Schedule Fall 2022-Spring 2023
Spring 2023
January 30, 2023
Jingwen Chen (U Chicago)
Time: 3:30 PM -4:30 PM, in person in VV901
Title: Mean curvature flows in the sphere via phase transitions.
Abstract: In this talk, we will discuss some solutions of the mean curvature flow (MCF) of surfaces in the 3-sphere. We will recall a generalized notion of MCF introduced by Brakke in the 70s, as well as its regularization by a parabolic partial differential equation arising in the theory of phase transitions. We will talk about some existence problems for this parabolic equation, and use them to construct MCFs that join minimal surfaces of low area in the 3-sphere, and some recent progress on the spaces of MCFs using Morse-Bott theory.
This is joint work with Pedro Gaspar (Pontificia Universidad Católica de Chile).
February 6, 2023
February 13, 2023
Trinh Tien Nguyen (UW Madison)
Format: In person, Time: 3:30-4:30PM.
Title: The inviscid limit of Navier-Stokes for domains with curved boundaries
Abstract: Understanding fluids with small viscosity is one of the most fundamental problems in mathematical fluid dynamics. The problem remains open in general for domains with curved boundaries, due to boundary layers near the boundary and large vorticity in the inviscid limit. We introduce the framework that captures precisely the pointwise behavior of the vorticity for the Navier-Stokes equations on domains with boundaries, under the no-slip boundary conditions. With a deep understanding of the linear problem with a nonlocal boundary condition of vorticity on the half-space, we show that the inviscid limit holds for the fully nonlinear Navier-Stokes equations if the initial data is locally analytic near the boundary, on a general bounded domain, or an exterior of a disk.
February 20, 2023
Ovidiu Avadanei UC Berkeley
Format: Format: In person in Room VV901, Time: 3:30-4:30PM.
Title: WELL-POSEDNESS FOR THE SURFACEQUASI-GEOSTROPHIC FRONT EQUATION
Abstract: We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation on the real line. Hunter-Shu-Zhang established well-posedness under a small data conditionas well as a convergence condition on an expansion of the equa-tion’s nonlinearity. In the present article, we establish uncondi-tional large data local well-posedness of the SQG front equationin the non-periodic case, while also improving the low regularitythreshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing bywave packet approach of Ifrim-Tataru.This is joint work with Albert Ai.
February 27, 2023
Yuxi Han (UW Madison)
Format: In person, Time: 3:30-4:30PM.
Title: Rate of Convergence in Periodic Homogenization for Convex Hamilton-Jacobi Equations with multiscales
Abstract: In reality, both macroscale and microscale variables are involved in various problems. But very often, we are only interested in macroscale behavior. Homogenization is the process of averaging out microscale behavior. In this talk, I will mainly focus on the rate of convergence in homogenization for convex Hamilton-Jacobi equations with multiscales. In particular, we show that for Cauchy problem, the rate of convergence is O(\sqrt{\epsilon}) and the power of \epsilon is optimal. This is a joint work with Jiwoong Jang.
March 6, 2023
Jinwoo Jang (Postech), Host: Chanwoo Kim
Format: In-person, Time: 3:30-4:30PM.
Title: Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus
Abstract: This talk deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The external magnetic potential well that we impose remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. This is a joint-work with Robert M. Strain and Tak Kwong Wong.
March 13, 2023
March 20, 2023
Format: , Time:
Title:
Abstract:
March 27, 2023
Matt Jacobs (Purdue). Host: Hung Tran.
Format: In person, Time: 3:30-4:30PM.
Title: Lagrangian solutions to the Porous Media Equation (and friends)
Abstract: Many works have been devoted to understanding and predicting the time evolution of a growing population of cells (bacterial colonies, tumors, etc...). At the macroscopic scale, cell growth is typically modeled through Porous Media type equations that describe the change in cell density. While these cell growth PDEs have been studied since the 70s, our understanding is far from complete, particularly in the case where there are several distinct cell populations.
An important open question is whether it is possible for two populations that were separated at initial time to become mixed during the flow. For instance, can tumor cells get mixed into healthy cell regions?
In this talk, I will show that it is possible to construct non-mixing solutions to these equations. The key is to construct the Lagrangian flow map along the pressure gradient generated by the Porous Media Equation. The main obstruction is the fact that the pressure gradient is not sufficiently regular to apply any generic theory for Lagrangian flows. To overcome this difficulty, we develop a new argument combining features of the Porous Media Equation with the quantitative Lagrangian flow theory of Crippa and De Lellis.
April 3, 2023
Zhihan Wang (Princeton). Host: Sigurd Angenent
Format: In person, Time: 3:30-4:30PM.
Title: Translating mean curvature flow with simple end.
Abstract: Translators are known as candidates of Type II blow-up model for mean curvature flows. Various examples of mean curvature flow translators have been constructed in the convex case and semi-graphical case, most of which have either infinite entropy or higher multiplicity asymptotics near infinity. In this talk, we shall present the construction of a new family of translators with prescribed end. This is based on joint work with Ao Sun.
April 10, 2023
Gi-Chan Bae (Seoul Nat. University), Host: Chanwoo Kim
Format: In person, Time: 3:30-4:30PM.
Title: Large amplitude solution of BGK model
Abstract: Bhatnagar–Gross–Krook (BGK) equation is a relaxation model of the Boltzmann equation which is widely used in place of the Boltzmann equation for the simulation of various kinetic flow problems. In this work, we study the asymptotic stability of the BGK model when the initial data is not necessarily close to global equilibrium pointwisely. The main difficulty of the BGK equation comes from the highly nonlinear structure of the relaxation operator. To overcomes this issue, we derive refined control of macroscopic fields to guarantee the system enters a quadratic nonlinear regime, in which the highly nonlinear perturbative term relaxes to essentially quadratic nonlinearity after a long but finite time. This is joint work with G.-H. Ko, D.-H. Lee and S.-B. Yun.
April 17, 2023
Jingrui Cheng (Stony Brook). Host: Misha Feldman.
Format: In person, Time: 3:30-4:30PM.
Title: Interior W^{2,p} estimates for complex Monge-Ampere equations
Abstract: The classical estimate by Caffarelli shows that a strictly convex solution to the real Monge-Ampere equations has W^{2,p} regularity if the right hand side is close to a constant. We partially generalize this result to the complex version, when the underlying solution is close to a smooth strictly plurisubharmonic function. The additional assumption we impose is related to the lack of Pogorelov type estimate in the complex case. The talk is based on joint work with Yulun Xu.
April 24, 2023
Ben Pineau (Berkeley)
Format: In person, Time: 3:30-4:30PM.
Title: Sharp Hadamard local well-posedness for the incompressible free boundary Euler equations, rough solutions, and continuation criterion
Abstract: We provide a complete, optimal local well-posedness theory for the free boundary incompressible Euler equations on a connected fluid domain, in $H^s$ based spaces. We establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data in low regularity Sobolev spaces; (ii) Enhanced uniqueness: our uniqueness result holds at the level of essentially the Lipschitz norm of the velocity and $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) stability bounds: Coupled with our uniqueness result are more general stability bounds; namely, we construct a nonlinear functional which measures in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions by making use of a newly constructed family of refined elliptic estimates; (v) The first proof of a sharp continuation criterion in pointwise norms, at the level of scaling, showing essentially that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$; (vi) A novel proof of construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is joint work in progress with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.
May 1, 2023
Chris Henderson (Arizona). Host: Chanwoo Kim
Time: 3:30 PM -4:30 PM, in person in VV901
Title:
Abstract:
Fall 2022
September 12, 2022
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
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
Format: , Time:
Title:
Abstract:
October 31, 2022
Yuan Gao (Purdue). Host: Hung Tran.
Format: in person, Time: 3:30pm-4:30pm VV 901
Title: A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures
Abstract: We will give a gentle introduction of weak KAM theory and then reinterpret Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. We will use one-dimensional irreversible diffusion process on torus to illustrate some essential concepts in the weak KAM theory such as the Peierls barrier, the projected Mather/Aubry/Mane sets. We provide alternative proofs for Freidlin-Wentzell's variational formulas for both self-consistent boundary data at each local attractors and for the rate function are formulated as the global adjustment for the boundary data and the local trimming from the lifted Peierls barriers. Based on this, we proved the rate function is a weak KAM solution to the corresponding stationary Hamilton-Jacobi equation satisfying the selected boundary data on projected Aubry set, which is also the maximal Lipschitz continuous viscosity solution. The rate function is the selected unique weak KAM solution and also serves as the global energy landscape of the original stochastic process. A probability interpretation of the global energy landscape from the weak KAM perspective will also be discussed.
November 7, 2022
Beomjun Choi (Postech)
Format: in person, Time: 3:30pm-4:30pm VV 901
Title: Liouville theorem for surfaces translating by powers of Gauss curvature
Abstract: We classify the entire solutions to degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^\beta$ on $\mathbb{R}^2$ for all $\beta<0$. The graphs of such solutions are the translating solitons to the flows by sub-affine-critical powers of Gauss curvature. In view of the Legendre transformation, this classifies the entire solutions to $\det D^2v = (1+|x|^2)^{-\beta}$ as well.
For the affine-critical-case $\det D^2u =1$, the celebrated result by Jorgens, Calabi and Pogorelov shows every solution must be a convex paraboloid and hence the level sets are homothetic ellipses. In our case, the level sets of given solution converge to a circle or a curve with $k$-fold symmetry for some $k>2$. These curves are closed shrinking curves to the curve shortening flows, classified by B. Andrews in 2003. Then we study the moduli space of solutions for each prescribed asymptotics. This is a joint work with K. Choi and S. Kim.
November 14, 2022
Adrian Tudorascu (West Virginia University). Host: Misha Feldman, Hung Tran.
Format: in person, Time: 3:30pm-4:30pm VV 901
Title: Sticky Particles with Sticky Boundary
Abstract: We study the pressureless Euler system in an arbitrary closed subset of the real line. The reflective boundary condition renders an ill-posed problem. Instead, we show that the sticky boundary condition is natural and yields a well-posed problem which is treated by means of sticky particles solutions, Lagrangian solutions and an appropriate (and natural) reflection principle.
November 21, 2022
Jason Murphy (Missouri S&T)
Format: online
Join Zoom Meeting https://uwmadison.zoom.us/j/94877483456?pwd=cG9Ec1dhb2ErcXhFVW1aN2hCYXRBUT09 Meeting ID: 948 7748 3456 Passcode: 303105
Time: 3:30 PM -4:30 PM
Title: Sharp scattering results for the 3d cubic NLS
Abstract: I will discuss several sharp scattering results for three-dimensional cubic nonlinear Schrödinger equations, including both the free NLS and the NLS with an external potential. After reviewing the proof of scattering below the mass/energy ground state threshold, I will discuss some work on scattering at the threshold for NLS with repulsive potentials. The talk will discuss joint works with B. Dodson; R. Killip, M. Visan, and J. Zheng; and C. Miao and J. Zheng.
November 28, 2022
No Seminar- Thanksgiving
December 5, 2022
James Rowan (UC Berkeley)
Time: 3:30 PM -4:30 PM
Title: Solitary waves for infinite depth gravity water waves with constant vorticity
Abstract: We show that solitary waves exist for pure gravity water waves in infinite depth in the presence of constant (nonzero) vorticity. The proof relies on the fact that this particular water-wave system is well-approximated by the Benjamin-Ono equation, which also allows a description of the profile of the solitary wave in terms of the Benjamin-Ono soliton. This is joint work with Lizhe Wan.
December 12, 2022
Calum Rickard UC Davis
Format: in-person in room VV901
Time: 3:00 PM -4:00 PM
Title: An infinite class of shocks for compressible Euler
Abstract: We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion. This is joint work with Sameer Iyer, Steve Shkoller and Vlad Vicol.
Schedule for Fall 2021-Spring 2022
PDE GA Seminar Schedule Fall 2021-Spring 2022
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
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.
March 14th, 2022.
March 21th, 2022.
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.
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.
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.
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
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
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Abstracts
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