Monday, July 14, 2025
・9:00 AM - 9:50 AM
Potential Analysis on Nonsmooth DomainsⅠ
Hiroaki Aikawa, Chubu University
Abstract: The Laplace and heat equations are classical and often considered well understood. However, many questions about the boundary behavior of solutions and supersolutions in general domains remain open. In this talk, we explore how domain complexity affects potential-theoretic properties, including the integrability of positive superharmonic functions, the Martin boundary, elliptic and parabolic boundary Harnack principles, intrinsic ultracontractivity, and related notions. To this end, we examine various nonsmooth domains, such as Lipschitz, NTA, uniform, inner uniform, John, Hölder, and $L^p$ domains, among others. The first lecture presents the main results, while the second provides an outline of the key techniques and proof strategies.
・10:00 AM – 10:50 AM
Potential Analysis on Nonsmooth DomainsⅡ
Hiroaki Aikawa, Chubu University
Abstract: The Laplace and heat equations are classical and often considered well understood. However, many questions about the boundary behavior of solutions and supersolutions in general domains remain open. In this talk, we explore how domain complexity affects potential-theoretic properties, including the integrability of positive superharmonic functions, the Martin boundary, elliptic and parabolic boundary Harnack principles, intrinsic ultracontractivity, and related notions. To this end, we examine various nonsmooth domains, such as Lipschitz, NTA, uniform, inner uniform, John, Hölder, and $L^p$ domains, among others. The first lecture presents the main results, while the second provides an outline of the key techniques and proof strategies.
・11:00 AM – 11:50 AM
To play around in a numerical sandbox to generate and illustrate potential theory conjectures
Torbjorn Lundh, Chalmers Inst of Technology, Sweden
Abstract: A useful classical method in analysis to think and generate new ideas is to sketch and doodle on paper or a black/white–board as an experimental sandbox to enhance our mental processes and to generate new ideas. I would like to exemplify a way to augment this classical method by using numerical methods, while hopefully still preserve our playfulness and creativity. As a first example, I would like to talk about some old, but still unpublished work, initiated with the collaboration, on the so-called 3G-inequality, with our here present delegate, professor Hiroaki Aikawa, resulting in “The 3G inequality for a uniformly John domain” (Kodai Mathematical Journal, 28(2): 209–219, 2005) extending an earlier result of Cranston, Fabes and Zhao: “Conditional gauge and potential theory for the Schrödinger operator” (Trans. Amer. Math. Soc. 307,1988). The second example of this numerical “sandbox technique” will be about an ill-posed free-boundary problem inspired from a biological process that could be seen as an inverse Heley-Shaw flow process. To conclude, the presentation will be focused how one could use high-levelnumerical tool boxes, such as Comsol Multiphysics, to play around to generate conjectures, to be later proven by classical analytic methods.
・2:00 PM – 2:50 PM
On the Second Order Energy Formulas of Quantum Gases
Xuwen Chen, University of Rochester, USA
Abstract: We survey the progress of the mathematical proof of 2nd order energy formulas of quantum gases like Lee-Huang-Yang, Gell-Mann-Brueckner, and Huang-Yang. We discuss the recent advancement of the proof of the Huang-Yang formula for Fermions.
( 2:50 PM – 3:30 PM Coffee Break )
・3:30 PM – 4:00 PM
The Helmholtz decomposition of a BMO type vector field in a domain
Zhongyang Gu, University of Tokyo, Japan
Abstract: The Helmholtz decomposition in the \(Lp\)-setting was well-studied for. However,it is not suitable to investigate this decomposition for the vector fields. In this talk, we will introduce a BMO type of vector fields in a domain, whose normal component to the boundary is well-controlled, and present its Helmholtz decomposition as a substitute theory for the setting. This talk is based on a series of joint works with Professor Yoshikazu Giga (The University of Tokyo).
・4:00 PM – 4:30 PM
James Warta, University of Missouri, USA
Abstract: The weak solutions to the parabolic Dirichlet problem on a domain whose boundary can be described locally the graph of a function that is Lipschitz with respect to the parabolic metric obey a Carleson measure estimate, then the corresponding parabolic measure on the boundary will belong to the Muckenhaupt class infinity. This improves the existing literature which places additional assumptions on the parabolic uniform rectifiability of the boundary or, equivalently, on the half-order time derivative.
Tuesday, July 15, 2025
・9:00 AM - 9:50 AM
On ruling out a class of type II blow-up scenarios in the hyper-dissipative Navier-Stokes equationsⅠ
Zoran Grujic, University of Alabama at Birmingham, USA
Abstract:
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system does not permit formation of singularities as long as the hyper-dissipation exponent, say beta, is greater or equal to 5/4. Recall that at 5/4 the system is in the critical regime — the energy level and the scaling-invariant levels coincide — while for beta greater than 5/4 the system is in the sub-critical regime. The question of global-in-time regularity in the super-critical regime, beta strictly between 1 and 5/4, has remained a fundamental open problem in mathematical fluid dynamics.
The main goal of the two lectures is to present a mathematical framework — built around a suitably defined scale of sparseness of the super-level sets of the components of the higher-order velocity derivatives — in which a class of `turbulent' blow-up scenarios can be ruled out as soon as the hyper-dissipation exponent is greater than 1. In particular, a class of type II generalized self-similar blow-ups is ruled out which — in turn — rules out approximately self-similar blow-ups, a prime candidate for singularity formation, in all 3D HD NS systems. This is a joint work with L. Xu.
・10:00 AM – 10:50 AM
Zoran Grujic, University of Alabama at Birmingham, USA
Abstract:
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system does not permit formation of singularities as long as the hyper-dissipation exponent, say beta, is greater or equal to 5/4. Recall that at 5/4 the system is in the critical regime — the energy level and the scaling-invariant levels coincide — while for beta greater than 5/4 the system is in the sub-critical regime. The question of global-in-time regularity in the super-critical regime, beta strictly between 1 and 5/4, has remained a fundamental open problem in mathematical fluid dynamics.
The main goal of the two lectures is to present a mathematical framework — built around a suitably defined scale of sparseness of the super-level sets of the components of the higher-order velocity derivatives — in which a class of `turbulent' blow-up scenarios can be ruled out as soon as the hyper-dissipation exponent is greater than 1. In particular, a class of type II generalized self-similar blow-ups is ruled out which — in turn — rules out approximately self-similar blow-ups, a prime candidate for singularity formation, in all 3D HD NS systems. This is a joint work with L. Xu.
・11:00 AM – 11:30 AM
Optimal regularity for kinetic equations in domains
Marvin Weidner, Universitat de Barcelona, Spain
Abstract: We study the smoothness of solutions to linear kinetic Fokker-Planck equations in domains with specular reflection condition on the boundary. While solutions are known to be smooth in the interior, their behavior near the boundary has remained open, even in the simplest case of Kolmogorov’s equation. In this talk I will report on a recent joint work with Xavier Ros-Oton, where we establish the optimal boundary regularity of solutions.
・2:00 PM – 2:50 PM
Edriss Titi, Texas A & M University, USA
Abstract: In these talks we will present rigorous analytical results concerning global regularity, in the viscous case, and finite-time singularity, in the inviscid case, for oceanic and atmospheric dynamics models. Moreover, we will also provide a rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. In addition, we will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation. Eventually, we will also present short-time well-posedness of solutions to the Hibler’s sea-ice model.
( 2:50 PM – 3:30 PM Coffee Break )
・3:30 PM – 4:20 PM
Edriss Titi, Texas A & M University, USA
Abstract: In these talks we will present rigorous analytical results concerning global regularity, in the viscous case, and finite-time singularity, in the inviscid case, for oceanic and atmospheric dynamics models. Moreover, we will also provide a rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations as the vanishing limit of the small aspect ratio of the depth to horizontal width. In addition, we will also show the global well-posedeness of the coupled three-dimensional viscous Primitive Equations with a micro-physics phase change moisture model for cloud formation. Eventually, we will also present short-time well-posedness of solutions to the Hibler’s sea-ice model.
・4:30 PM – 5:00 PM
Lyapunov stability and exponential phase-locking of Schrödinger-Lohe oscillators
David Reynolds, Universidad de Granada, Spain
Abstract: In this talk based off of joint works with Paolo Antonelli (GSSI) we ill discuss some basics of synchronization dynamics. Then we will introduce the Schrödinger-Lohe model for quantum synchronization. The model is described by a system of Schrödinger equations, coupled through nonlinear, non-Hamiltonian interactions that drive the system towards phase synchronization. The model can be viewed as a quantum generalization of the famous Kuramoto model of phase-synchronization. Despite enjoying similar structural qualities, until recently stability and conver-ence to phase-locked state for nonidentical oscillators has been elusive. e present such stability and convergence results which brings the state f the art for the Schrödinger-Lohe model closer to that of other models within the Kuramoto family. Keywords: emergence, quantum synchronization, Schrödinger-Lohe model.
Wednesday, July 16, 2025
・9:00 AM - 9:50 AM
Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan
Abstract: This talk will address the major problem in the Analysis of PDEs on the nature of singularities reflecting the natural phenomena. I will present my solution of the Kolmogorov's Problem (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Brownian motion, and criteria for thinness in minimal-fine topology. The talk will end with the description of some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs.
・10:00 AM – 10:50 AM
Ugur G. Abdulla, Okinawa Institute of Science and Technology, Japan
Abstract: In this talk I will sketch the proof of the new Wiener-type criterion for the removability of the fundamental singularity and, equivalently, the unique solvability of the singular Dirichlet problem for the heat equation. This work fully characterizes the removability of non-isolated boundary singularities through the fine-topological thinness of the exterior set near the singularity point. A significant tool in this characterization is the new concept of h-capacity of Borel sets, which measures thinness and establishes singularity behavior via the divergence of weighted sums of h-capacities within nested shells.
・11:00 AM – 11:30 AM
Regularity of $\infty$ for a class of linear degenerate elliptic equations
Denis Brazke, Okinawa Institute of Science and Technology, Japan
Abstract: In this talk, I present recent progress on the topic of well-posedness of the Dirichlet problem for linear degenerate elliptic equations in arbitrary open sets in dimension $n \geq 2$. More specifically, the corresponding weight of the ellipticity is assumed to behave like a polynomial at infinity with exponent $\gamma > 2 – n$. After introducing the concept of regularity of $\infty$ and the harmonic measure at $\infty$, I discuss its connection to well-posedness of the Dirichlet problems and a weighted Wiener criterion at $\infty$.
This is joint work with Prof. Dr. Ugur Abdulla.
・2:00 PM – 9:00 PM
Excursion (Asmui Hikes)
Thursday, July 17, 2025
・9:00 AM - 9:50 AM
Nonlinear Potentials in PDE: from uniformly to nonuniformly elliptic problemsⅠ
Giuseppe Mingione, University of Parma, Italy
Abstract: In these two talks I will summarize a series of recent results in regularity theory of elliptic PDEs, where nonlinear potentials play a key role in order to get sharp a priori regularity estimates. In particular, in the first talk I will concentrate on uniformly, yet degenerate elliptic problems. In this situation it is possible to derive neat, and optimal, potential estimates for solutions. These in turn imply sharp regularity properties in terms of given data.
・10:00 AM – 10:50 AM
Nonlinear Potentials in PDE: from uniformly to nonuniformly elliptic problemsⅡ
Giuseppe Mingione, University of Parma, Italy
Abstract: In these two talks I will summarize a series of recent results in regularity theory of elliptic PDEs, where nonlinear potentials play a key role in order to get sharp a priori regularity estimates. In particular, in the second talk I will concentrate on nonuniformly elliptic problems, showing a few new techniques eventually leading to the proof of Schauder estimates. These involve a delicate use of a general class of nonlinear potentials.
・11:00 AM – 11:50 AM
The Dirichlet problem as the boundary of the Poisson problem
Bruno Poggi, University of Pittsburgh, USA
Abstract: We review certain classical quantitative estimates (known as non-tangential maximal function estimates) for the solutions to the Dirichlet boundary value problem for the Laplace equation in a smooth domain in Euclidean space, when the boundary data lies in an $L^p$ space, $p>1$. A natural question that arises is: what might an analogous estimate for the inhomogeneous Poisson problem look like? We will answer this question precisely, and in so doing, we will unravel deep and new connections between the solvability of the (homogeneous) Dirichlet problem for the Laplace equation with data in $L^p$ and the solvability of the (inhomogeneous) Poisson problem for the Laplace equation with data in certain Carleson spaces. We employ this theory to solve a 20-year-old problem in the area, to give new characterizations and a new local T1-type theorem for the solvability of the Dirichlet problem under consideration, and to furnish a bridge to the mathematical physics theory of the Filoche-Mayboroda landscape function. The new results are the product of joint work with Mihalis Mourgoglou and Xavier Tolsa.
・2:00 PM - 2:50 PM
On well-posedness of $s$-Schrödinger maps
Armin Schikorra, University of Pittsburgh, USA
Abstract: I am going to present recent progress on well-posedness of a nonlinear Schrödinger system with loss of derivatives that is a model equation for the $s$-Schrödinger map system \(\partial_t u = u \wedge (-\Delta)^s u\) - which for $s = 1/2$ is the halfwave map equation, for $s = 1$ it is the Schrödinger map equation. We consider the case \(s \in (1/2,1)\).
Joint work with Ahmed Dughayshim and Silvino Reyes-Farina.
( 2:50 PM – 3:30 PM Coffee Break )
・3:30 PM – 4:20 PM
Jin Feng, University of Kansas, USA
Abstract: We take a variational approach to understand hydrodynamic limit of (global) action-mininizing Lagrangian collective dynamics, of weakly interacting particles. Through the study of limit theorems on Hamilton-Jacobi equation in space of probability measures, we derive a hydrodynamic limit effective Hamiltonian and corresponding variational problem. We make extensive use of recent theories of optimal transport and first order analysis in Alexandrov metric spaces (for understanding the PDEs involved), and the weak KAM theory in finite dimensions (for establishing micro-canonical ensemble).
・4:30 PM – 5:00 PM
Recent advances in nonlocal potential theory
Minhyun Kim, Hanyang University, South Korea
Abstract: Nonlocal potential theory is the study of $L$-harmonic functions with respect to nonlocal operators $L$ modeled on the fractional Laplacian. In this talk, I will present recent results on boundary behavior of $L$-harmonic functions. The main topics include boundary regularity, Wiener criterion, Green function estimates and classification of regular boundary points. This talk is based on joint works with Anders Björn, Jana Björn, Ki-Ahm Lee, Se-Chan Lee, and Marvin Weidner.
Friday, July 18, 2025
・9:00 AM - 9:50 AM
Coupling and Ishii-Lions Methods for Tug-of-War Stochastic Games with NoiseⅠ
Juan Manfredi, University of Pittsburgh, USA
Abstract: We review the theory of viscosity solutions to non-linear elliptic partial differential equations, including the Theorem of Sums. We then provide a detailed exposition of two distinct regularity methodologies and explore their interrelation. We examine the coupling method within the framework of tug-of-war stochastic games augmented by noise and address the regularity of viscosity solutions to the $p$-Laplace equation using the Ishii-Lions method. In the first talk, we present relevant definitions, examples, theorems,and further research directions. The details of the proofs will be discussed in the second talk.
・10:00 AM – 10:50 AM
Coupling and Ishii-Lions Methods for Tug-of-War Stochastic Games with NoiseⅡ
Juan Manfredi, University of Pittsburgh, USA
Abstract: We review the theory of viscosity solutions to non-linear elliptic partial differential equations, including the Theorem of Sums. We then provide a detailed exposition of two distinct regularity methodologies and explore their interrelation. We examine the coupling method within the framework of tug-of-war stochastic games augmented by noise and address the regularity of viscosity solutions to the $p$-Laplace equation using the Ishii-Lions method. In the first talk, we present relevant definitions, examples, theorems,and further research directions. The details of the proofs will be discussed in the second talk.
・11:00 AM – 11:50 AM
Inside out without effort: The geometry of isometric band eversions
Eliot Fried, Okinawa Institute of Science and Technology, Japan
Abstract: We present an analytical framework for describing continuous, periodic everting motions of orientable and nonorientable bands, including configurations exhibiting twists and knots. These motions arise as traveling wave solutions of a nonlinear integro-partial-differential equation governing the evolution of the generatrix of the band . At each instant of an eversion cycle, the configuration is constructed through periodic or antiperiodic extensions of the initial data, depending on whether the band is orientable or nonorientable, respectively. The intrinsic metric and the total elastic bending energy both remain constant throughout these motions, which constitute a novel class of shape-preserving, nonrigid transformations. Unlike classical eversions in nonlinear elasticity, which inevitably involve stretching and energy input, these proceed without such cost. We construct explicit traveling wave solutions and characterize their spatial and temporal periodicity, thereby contributing new insights into the geometry of ruled surfaces with nontrivial topology and their implications for soft robotic and adaptive structural design. This is joint work with Vikash Chaurasia.
・2:00 PM – 2:50 PM
Optimal Liouville theorems for conformally invariant PDEs
Zongyuan Li, City University of Hong Kong, Hong Kong
Abstract: The celebrated result of Caffarelli, Gidas, and Spruck (1989) classified all nonnegative solutions to a class of semilinear elliptic PDEs, establishing a cornerstone Liouville-type theorem. In this talk, we present an optimal generalization to the fully nonlinear, conformally invariant setting. Time permitting, we will also discuss some applications in conformal geometry. This is joint work with B. Z. Chu and Y. Y. Li (Rutgers).
( 2:50 PM – 3:30 PM Coffee Break )
・3:30 PM – 4:20 PM
Isoperimetric inequality for multiply winding curves
James McCoy, University of Newcastle, Australia
Abstract: We’ll discuss some old and new results for the isoperimetric inequality in the plane, for closed curves of positive integer winding number. In particular, I will outline some work in progress with Yong Wei and Glen Wheeler on a new nonlinear fourth order parabolic curvature flow that can be used to prove an isoperimetric inequality for multiply winding curves of sufficiently small oscillation of curvature. Time permitting, I will outline how this flow can also be used to prove a similar result for embedded curves in the sphere and multiply-winding curves in hyperbolic space.