# Difference between revisions of "PDE Geometric Analysis seminar"

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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 ! | 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 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''' | ||

+ | |||

+ | [[No seminar.]] | ||

+ | |||

+ | 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 ''' | ||

+ | |||

+ | [[Calumn Rickard ]] UC Davis | ||

+ | |||

+ | Format: , Time: | ||

+ | |||

+ | Title: | ||

+ | |||

+ | Abstract: | ||

+ | |||

+ | |||

+ | |||

+ | '''March 27, 2023''' | ||

+ | |||

+ | [[Yuan Gao|Matt Jacobs]] (Purdue). Host: Hung Tran. | ||

+ | |||

+ | Format: , Time: | ||

+ | |||

+ | Title: | ||

+ | |||

+ | Abstract: | ||

+ | |||

+ | |||

+ | |||

===[[Fall 2021-Spring 2022 | Schedule for Fall 2021-Spring 2022]]=== | ===[[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—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 [[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. | ||

+ | =Fall 2021= | ||

'''September 20th, 2021.''' | '''September 20th, 2021.''' | ||

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'''September 27th, 2021.''' | '''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). | ||

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[[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM | [[Antoine Remind-Tiedrez]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM | ||

− | Title: | + | 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 11th, 2021.''' | ||

− | + | [[No seminar]] | |

− | |||

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[[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM | [[Wojciech Ozanski]] (USC); Format: online seminar via Zoom, Room:--, Time: 3:30PM-4:30PM | ||

− | Title: | + | Title: Well-posedness of logarithmic spiral vortex sheets. |

− | Abstract: | + | 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.''' | '''November 1th, 2021.''' | ||

− | [[ | + | [[Lizhe Wan]] (UW Madison); Format: in-person seminar, Room: 901, Time: 3:30PM-4:30PM |

− | Title: | + | Title: The Benjamin-Ono approximation for 2D gravity water waves with constant vorticity |

− | Abstract: | + | 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. |

Line 76: | Line 459: | ||

'''November 8th, 2021.''' | '''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.''' | '''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 ]] | ||

## Latest revision as of 21:56, 29 November 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

### Previous PDE/GA seminars

### Schedule for Fall 2022-Spring 2023

## PDE GA Seminar Schedule Fall 2022-Spring 2023

# 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 **

Calumn Rickard UC Davis

Format: , Time:

Title:

Abstract:

**March 27, 2023**

Matt Jacobs (Purdue). Host: Hung Tran.

Format: , Time:

Title:

Abstract:

### 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

Title:

Abstract: