PDE-Applied Math Archives for Fall 2023 to Spring 2024
Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian
When: Thu, September 29, 2022 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Mickaël Latocca (UMD)
https://www.math.ens.psl.eu/~latocca/index.html
Set minimizers of attractive-repulsive energies
When: Thu, October 27, 2022 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Ihsan A. Topaloglu (VCU)
Abstract: In this talk I will consider a class of attractive-repulsive energies, given by the sum of two nonlocal interactions with power-law kernels, defined over sets with fixed measure. After a review of the literature on this minimization problem, I will focus on the issue of the stability of the ball, in the sense of the positivity of the second variation of the energy with respect to smooth perturbations of the boundary of the ball. For a certain choice of interaction kernels, I will present a characterization of the range of masses for which the second variation is positive definite (large masses) or negative definite (small masses). Moreover, I will discuss the connection between the stability of the ball and its local/global minimality. Finally, time permitting, I will show that for certain interactions set minimizers exist even in the small mass regime, and they are not radially symmetric.
Physics-informed neural networks for self-similar blow-up solutions
When: Thu, November 3, 2022 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Yongji Wang
Abstract: One of the most challenging open questions in mathematical fluid dynamics is whether an inviscid incompressible fluid, described by the 3-dimensional Euler equations, with initially smooth velocity and finite energy can develop singularities in finite time. This long-standing open problem is closely related to one of the seven Millennium Prize Problems which considers the Navier-Stokes equations, the viscous analogue to the Euler equations. In this talk, I will describe why and how the physics-informed neural networks (PINNs) can be a robust and universal tool to find the smooth self-similar blow-up solution for various fluid equations, from the simple 1-D burgers equation to the 3-D Euler equations in the presence of a cylindrical boundary. To the best of our knowledge, the latter represents the first example of a truly 2-D or higher dimensional backwards self-similar solution. This sheds new light to the century-old mystery of capital importance in the field of mathematical fluid dynamics.
Probabilistic global well-posedness for NLS-type equations
When: Thu, November 10, 2022 - 3:30amWhere: Kirwan Hall 3206
Speaker: Mouhamadou Sy (Johns Hopkins)
Abstract: We consider the defocusing nonlinear Schrodinger equations and their fractional versions. On bounded domains, these equations are not mathematically very well understood, in particular when the nonlinearities are supercritical. We construct global solutions via an invariant measure method and discuss long-time behavior of the solutions. The main challenge is the supercriticality which does not allow the standard invariant measure arguments.
On unique continuation at the boundary for elliptic operators
When: Thu, November 17, 2022 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Shiferaw Berhanu (UMD)
Abstract: In 1993, M.S. Baouendi and Linda Rothschild proved the following result for the Laplace operator: Let B+ be a half ball in the upper half space in R^n, u harmonic in B+, and u(x′, 0) ≥ 0 for x′ in a neighborhood of the origin on the flat piece of ∂B+. If u vanishes to infinite order at the origin in the sense that u(x) = O(|x|^N) for all N, then u ≡ 0. They conjectured that a similar result holds for any second order elliptic operator with real analytic coefficients. We will present our positive solution of the conjecture and an extension to operators of any order. Our results have applications to unique continuation for CR functions (in several complex variables) which was the original inspiration for Baouendi and Rothschild.
The many, elaborate wrinkle patterns of confined elastic shells
When: Thu, December 8, 2022 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Ian Tobasco https://www.math.uic.edu/persisting_utilities/people/profile?netid=itobasco
Abstract: Abstract:A basic fact of geometry is that there are no length-preserving smooth maps from a spherical cap into a plane. But what happens if you try to press a curved elastic shell into a plane anyways? It wrinkles along a remarkable pattern of peaks and troughs, the arrangement of which is sometimes random, sometimes not depending on the shell. After a brief introduction to the mathematics of thin elastic sheets and shells, this talk will focus on a new set of simple, geometric rules we have discovered for predicting wrinkles driven by confinement. These rules are the latest output from an ongoing study of elastic confinement using the tools of Gamma-convergence and convex analysis. The asymptotic expansions they encode reveal a beautiful and unexpected connection between opposite curvatures — apparently, surfaces with positive or negative Gaussian curvatures can be paired according to the way that they wrinkle when confined. Our predictions match the results of numerous experiments and simulations done with Eleni Katifori (U. Penn) and Joey Paulsen (Syracuse). Underlying their analysis is a certain class of interpolation inequalities of Gagliardo-Nirenberg type, whose best prefactors encode the optimal patterns.
Logarithmic spirals in 2d Perfect fluids
When: Thu, February 2, 2023 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Ayman Rimah (Duke University) - https://aymanrimah.github.io
Abstract: In this talk I discuss recent results in collaboration with In-Jee Jeong from Seoul National University. We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the circle. We show that this system is locally well-posed in $L^p, p\geq 1$ as well as for atomic measures, that is vortex logarithmic spiral sheets. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. Within symmetry we show that the vortex sheet limit holds locally in time. We give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and do occur
Phase transitions and log Sobolev inequalities
When: Thu, March 16, 2023 - 3:30pmWhere: Kirwan Hall MTH1311
Speaker: Matias Delgadino (UT Austin)
Abstract: In this talk, we will study the mean field limit of weakly
interacting diffusions for confining and interaction potentials that
are not necessarily convex. We explore the relationship between the
large N limit of the constant in the logarithmic Sobolev inequality
(LSI) for the N-particle system, and the presence or absence of phase
transitions for the mean field limit. The non-degeneracy of the LSI
constant will be shown to have far reaching consequences, especially
in the context of uniform-in-time propagation of chaos and the
behaviour of equilibrium fluctuations. This will be done by employing
techniques from the theory of gradient flows in the 2-Wasserstein
distance, specifically the Riemannian calculus on the space of
probability measures.
Interior regularity for stationary two-dimensional multivalued maps
When: Thu, March 30, 2023 - 3:30pmWhere: Kirwan Hall MTH1311
Speaker: Luca Spolaor https://mathweb.ucsd.edu/~lspolaor/
Abstract: $Q$-valued maps minimizing a suitably defined Dirichlet energy were introduce by Almgren in his proof of the optimal regularity of area minimizing currents in any dimension and codimension. In this talk I will discuss the extension of Almgren's result to stationary $Q$-valued maps in dimension $2$. This is joint work with Jonas Hirsch (Leipzig).
Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit
When: Thu, April 13, 2023 - 3:00pmWhere: Kirwan Hall MTH1311
Speaker: Robin Neumayer
Abstract: The Alt-Caffarelli-Friedman (ACF) monotonicity formula is an important tool in the study of free boundary problems. More generally, given any pair of nonnegative subharmonic functions with disjoint positivity sets, the ACF formula provides information about the interface between the supports. In this talk we’ll show that on the portion of the interface where the ACF formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set, and that the two functions have a unique blowup at $H^{n-1}$ almost every such point. This talk is based on joint work with Mark Allen and Dennis Kriventsov.
Harmonic maps with free boundary and beyond
When: Thu, April 20, 2023 - 3:30pmWhere: Kirwan Hall MTH1311
Speaker: Yannick Sire (John Hopkins)
Abstract: I will introduce a new heat flow for harmonic maps with free boundary. After giving some motivations to study such maps in relation with extremal metrics in spectral geometry, I will construct weak solutions for the flow and derive their partial regularity. The introduction of this new flow is motivated by the so-called half-harmonic maps introduced by Da Lio and Riviere, which provide a new approach to the old topic of harmonic maps with free boundary. I will also state some open problems and possible generalizations.
Dynamics of multi-solitons to Klein-Gordon equations
When: Thu, April 27, 2023 - 2:00pmWhere: Kirwan Hall MTH1311
Speaker: Gong Chen (Georgia Tech) https://sites.google.com/site/cg66math/home
Abstract: I will report my recent joint work with Jacek Jendrej on muti-solitons to the Klein-Gordon equations including their asymptotic stability and classification.
Regularity results for area minimizing $m$-currents in $\mathbb{R}^{m+n}$
When: Thu, May 4, 2023 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Reinaldo Resende (University of São Paulo) - https://www.ime.usp.br/~resende/
Abstract: The focus will be the study of the regularity of currents that solve the so-called oriented Plateau's problem. A description of the results available in the literature will be made, focusing on the differences between the approaches to treat cases of codimension 1 and higher codimensions. We will walk through De Giorgi and Almgren's strategies to prove regularity.
The desingularization of small moving corners for the Muskat equation
When: Thu, May 11, 2023 - 3:30pmWhere: Kirwan Hall 1311
Speaker: Susanna Haziot (Brown University) https://susanna-haziot.com/
Abstract: The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez (Universidad de Sevilla), Javier Gomez-Serrano (Brown University) and Benoit Pausader (Brown University).