Recent Advances in Potential Theory and Partial Differential Equations

After Newton’s discovery of the universal law of gravitation and Gauss’s discovery of the flux theorem for gravity, the main progress in the classical physics was the development of the Potential Theory, which provides the mathematical representation of gravitational fields. Modern Potential Theory is a field of Pure Mathematics in the cross-section of Analysis, Partial Differential Equations (PDE) and Probability Theory, and plays a crucial role for the study of many different phenomena in fluid dynamics, electrostatics and magnetism, quantum mechanics and probability theory. The conference will outline the major current developments in potential theory of elliptic and parabolic PDEs, nonlinear PDE systems in fluid mechanics, including the regularity of weak solutions, Wiener-type criteria for the boundary regularity, regularity of the point at infinity and its probabilistic, measure-theoretical and topological consequences, criteria for the removability of singularities of PDEs representing natural phenomena, singularities of the Navier-Stokes equations, free boundary problems, asymptotic laws for the diffusion processes, conditional Brownian processes, fine and minimal-fine topology and other related topics.

 

PLENARY SPEAKERS: 

 

Prof. Ugur G. Abdulla | OIST, Japan

Prof. Hiroaki Aikawa | Chubu University, Japan

Prof. Edriss Titi | Texas A & M University, USA

Prof. Giuseppe Mingione | University of Parma, Italy

Prof. Juan Manfredi | University of Pittsburgh, USA

Prof. Zoran Grujic | University of Alabama at Birmingham, USA

Prof. Zhou-ping Xin | Chinese University of Hong Kong, Hong Kong

 

Schedule

 

Monday, July 14

・9：00 - 9：50   Hiroaki Aikawa, Chubu University    

Title: Potential Analysis on Nonsmooth DomainsⅠ

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, H\"older, 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 – 10:50   Hiroaki Aikawa, Chubu University　 

Title: Potential Analysis on Nonsmooth DomainsⅡ

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, H\"older, and LpLp 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 – 11:50   Zhou-ping Xin, Chinese University of Hong Kong   Title and Abstract: TBA

・14:00 – 14:50   Armin Shikorra, University of Pittsburgh   

Title: On well-posedness of s-Schroedinger maps

Abstract: I am going to present recent progress on well-posedness of a nonlinear Schroedinger system with loss of derivatives that is a model equation for the s-Schroedinger map system \partial u = u∧ (—∆)ˢu – which for s = 1/2 is the halfwave map equation for s = 1 it is the Schroedinger map equation. We consider the case s ∈ (1/2,1).

Joint work with Ahmed Dughayshim and Silvino Reyes-Farina.

 

・14:50 – 15:30 Coffee Break  

・15:30 – 16:00    

 

・16:00 – 16:30   Zhongyang Gu, University of Tokyo   

Title: The Helmholtz decomposition of a BM O type vector field in a domain

Abstract: The Helmholtz decomposition in the Lp-setting was well-studied for 1 < p < ∞. However,it is not suitable to investigate this decomposition for the L∞ vector fields. In this talk, we will introduce a BM O 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 L∞-setting. This talk is based on a series of joint works with Professor Yoshikazu Giga (The University of Tokyo).

 

Tuesday, July 15

・9：00 - 9：50  Edriss Titi, Texas A & M University    

Title: Mathematical Analysis of Atmospheric and Oceanic Dynamics Models: Cloud Formation and Sea-ice Models Ⅰ

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.

・10:00 – 10:50   Edriss Titi, Texas A & M University   

Title: Mathematical Analysis of Atmospheric and Oceanic Dynamics Models: Cloud Formation and Sea-ice Models Ⅱ

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.

・11:00 – 11:50   Zoran Grujic, University of Alabama at Birmingham   Title and Abstract: TBA

・14:00 – 14:50    Xuwen Chen, University of Rochester    Title and Abstract: TBA

・14:50 – 15:30 Coffee Break

・15:30 – 16:00    Marvin Weidner, Universitat de Barcelona, Spain   

Title: Optimal regularity for nonlocal free boundary problems

Abstract: Free boundary problems have been a central topic of research in PDE theory for the last fifty years. An increasingly prominent class is that of nonlocal free boundary problems, which naturally arises in models where long range interactions need to be taken into account.
In this talk, I will present a recent result on the optimal regularity of solutions to obstacle problems for general nonlocal operators. Interestingly, our approach draws on insights from a seemingly distinct problem, the nonlocal one-phase free boundary problem. This talk is based on joint works with Xavier Ros-Oton.

・16:00 – 16:30   David Reynolds, Universidad de Granada, Spain    

Title: Lyapunov stability and exponential phase-locking of Schro¨dinger-Lohe oscillators

Abstract:  In this talk based off of joint works with Paolo Antonelli (GSSI) we ill discuss some basics of synchronization dynamics. Then we will in-troduce the Schro¨dinger-Lohe model for quantum synchronization. The model is described by a system of Schro¨dinger equations, coupled through nonlinear, non-Hamiltonian interactions that drive the system towards hase synchronization. The model can be viewed as a quantum general-ization 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 Schro¨dinger-Lohe model closer to that of other models within the Kuramoto family. Keywords: emergence, quantum synchronization, Schro¨dinger-Lohe model.

 

Wednesday, July 16

・9：00 - 9：50  Ugur G. Abdulla, OIST, Japan    Title and Abstract: TBA

・10:00 – 10:50   Ugur G. Abdulla, OIST, Japan      Title and Abstract: TBA

・11:00 – 11:50   Irina Mitrea, Temple University, USA   Title and Abstract: TBA

・ 14:00 – 21:00 Field trip

 

Thursday, July 17

・9：00 - 9：50   Giuseppe Mingione, University of Parma, Italy     Title and Abstract: TBA

・10:00 – 10:50   Giuseppe Mingione, University of Parma, Italy      Title and Abstract: TBA

・11:00 – 11:50   Bruno Poggi, University of Pittsburgh, USA     Title and Abstract: TBA

・14:00 – 14:50    Zongyuan Li, City University of Hong Kong    Title and Abstract: TBA

・14:50 – 15:30 Coffee Break 

・15:30 – 16:00    Minhyun Kim, Hanyang University, South Korea   Title and Abstract: TBA

・16:00 – 16:30    James Warta, University of Missouri, USA    

Title: Carleson Measure Estimates Imply the Parabolic Measure is Muckenhaupt Infinity in the Case of a Graph Domain That's Lipschitz With Respect to the Parabolic Metric

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.

 

Friday, July 18

・9：00 - 9：50   Juan Manfredi, University of Pittsburgh, USA     Title and Abstract: TBA

・10:00 – 10:50   Juan Manfredi, University of Pittsburgh, USA      Title and Abstract: TBA

・11:00 – 11:50   Torbjorn Lundh, Chalmers Inst of Technology, Sweden    Title and Abstract: TBA

・14:00 – 14:50   Eliot Fried, OIST, Japan    Title and Abstract: TBA

・14:50 – 15:30 Coffee Break

・15:30 – 16:00    James McCoy, University of Newcastle, Australia    

 

Title: Isoperimetric inequality for multiply winding curves

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.

・16:00 – 16:30   Abderrahmane Lakhdari, Tunisia    Title and Abstract: TBA