Titles and Abstracts

Asymptotic properties of some non-coercive Hamiltonians.

Martino Bardi (Università di Padova, Italy)

Several interesting asymptotic properties of Hamilton-Jacobi equations are based on the so-called critical value of the Hamiltonian $H(x,p)$$H(x,p)$ and on the associated critical stationary H-J equation. In particular, the long time behaviour of evolutive H-J equations is described in terms of the critical value and a critical solution, and so is the homogenisation of H-J equations with highly oscillating ingredients. The theory was pioneered by Lions, Papanicolaou and Varadhan and by A. Fathi, and it has applications to ergodic control and to dynamical systems, the so-called weak KAM theory. Most of the known result assume the coercivity of the Hamiltonian in the moment variables $p$$p$, and interpret the critical equation in the sense of continuous viscosity solutions.

I will present some recent improvements holding for non-coercive Hamiltonians arising from the optimal control of affine systems, possibly with an uncontrolled drift term. Different from the previous theory, I use in a crucial way viscosity solutions that can be discontinuous. I will also mention some partial results for convex-concave Hamiltonians arising from 2-person, 0-sum differential games.

Finally, I will briefly describe some similar problems and results for the first order systems of backward-forward PDEs arising in the theory of deterministic Mean Field Games.

Singular Set in Optimal Transportation

Shibing Chen (University of Science and Technology of China, China)

It is well known that the optimal transport problem can develop singularities when the target domain is non-convex. In this talk, I will discuss some results concerning the regularity of the singular set in optimal transportation.

On the convergence rate for vanishing viscosity approximation of HJ equations

Marco Cirant (Università di Padova, Italy)

We discuss the sharp rate of convergence in sup-norm of the vanishing viscosity limit of Hamilton-Jacobi equations with convex Hamiltonians, with periodic data. We focus first on uniformly convex Hamiltonians, then on strictly convex Hamiltonians. The results are based on the nonlinear adjoint method. Joint work with A. Goffi (University of Firenze)

Stochastic Homogenization of viscous HJ equations in 1d

Andrea Davini (Università di Roma, La Sapienza, Italy)

In this talk I will present some new results I have recently obtained about homogenization of viscous Hamilton-Jacobi equations in dimension one in stationary ergodic environments with nonconvex Hamiltonians. In the non-degenerate case, i.e., when the diffusion coefficient is strictly positive, homogenization is established for superlinear Hamiltonians of fairly general type. This closes a long standing question. When, on the other hand, the diffusion coefficient degenerates, meaning that it is zero at some points or on some regions of the real line, homogenization is proved for Hamiltonians that are additionally assumed quasiconvex in the momentum variable. Furthermore, the effective Hamiltonian is shown to be quasiconvex. This latter result is new even in the periodic setting, despite homogenization being known for quite some time.

What Do We Know about Stochastic Hamiltonian Dynamics?

Jinqiao Duan (Great Bay University, China)

Dynamical systems are often under the influence of random fluctuations. In particular, stochastic Hamiltonian dynamical systems have attracted a lot of attention recently. The interactions between Hamiltonian structures and uncertainty are leading to fascinating dynamical phenomena.

The speaker will overview recent advances or research issues in stochastic Hamiltonian dynamics, including impact of noise on Hamiltonian structure， symplectic vs contact structures, and stochastic Hamilton-Jacobi equation.

Trading linearity for ellipticity - a novel approach to global Lorentzian geometry

Nicola Gigli (SISSA, Italy)

The concepts of Sobolev functions, elliptic operators and Banach spaces are central in modern geometric analysis. In the setting of Lorentzian geometry, however, unless one restricts the attention to Cauchy hypersurfaces these do not have a clear analogue, due to the signature of the metric tensor. Aim of the talk is to discuss some recent observations in this direction centered around the fact that for $p<1$$p<1$ the $p$$p$-D’Alambertian is elliptic on the space of time functions.

The talk is mostly based on joint project with Beran, Braun, Calisti, McCann, Ohanyan, Rott, Saemann.

Monotonicity Methods for Mean Field Games: A Functional Analytic Perspective

Diogo Gomes (King Abdullah University of Science and Technology, Saudi Arabia)

This presentation explores mean field games (MFGs) through the lens of functional analysis, focusing on the role of monotonicity methods in understanding their properties and deriving solutions.

Monotonicity operators emerge as a central tool in our analysis. We establish the connection between monotone operators and variational inequalities, showcasing how the latter offers a flexible framework for addressing situations where traditional solutions may not exist.

Building on this foundation, we extend our discussion to the Banach space setting, examining monotone operators between a Banach space and its dual. We present existence theorems and regularization methods tailored to this context. We conclude by exploring the concept of weak-strong uniqueness, which establishes conditions under which weak and strong solutions of MFGs coincide.

On the geometry of Wasserstein barycenter

Bangxian Han (Shandong University, China)

Wasserstein barycenter problem, which arises from the optimal transport theory, has got a lot of attention in various fields in the last decade. I will introduce some recent progress about Wasserstein barycenter problem in the setting of extended metric measure spaces. I will focus on the key roles of curvature and differential structure in the study of this problem. Based on joint works with Dengyu Liu and Zhuonan Zhu.

Variational construction of singular characteristics and propagation of singularities

Jiahui Hong (Nanjing University of Aeronautics and Astronautics, China)

In this talk, we begin with the problem of propagation of singularities for Hamilton-Jacobi equation. We introduce a novel theory of maximal slope curves for any pair $(\varphi ,H)$$(\phi,H)$ with $\varphi$$\phi$ a semiconcave function and $H$$H$ a Hamiltonian. We prove the existence and stability of such maximal slope curves and prove that these curves are exactly broken characteristics. Moreover, we give a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities. This is based on joint work with Piermarco Cannarsa, Wei Cheng and Kaizhi Wang.

Dynamics of action minimizing orbits of characteristic system for Hamilton-Jacobi equations

Liang Jin (Nanjing University of Science and Technology, China)

Let $M$$M$ be a closed Riemannian manifold. In recent years, rich achievements on the study of Hamilton-Jacobi equations $H(x,u,{\partial }_{x}u)=0$$H(x,u,\partial_xu)=0$ and ${\partial }_{t}u+H(x,u,{\partial }_{x}u)=0$$\partial_t u+H(x,u,\partial_xu)=0$ defined on $M$$M$ have been obtained via knowledge on their characteristic systems. In this talk, we will present some results in the opposite direction, namely, drawing conclusions on the dynamics of certain action minimizing characteristics from the knowledge of the construction and asymptotic behavior of the solutions to the Hamilton-Jacobi equations. The difference between the dynamics of such characteristic system and classical Hamiltonian systems as well as some applications will be explained. This based on joint works with Jun Yan (Fudan Univesity) and Kai Zhao (Tongji University).

On the stochastic homogenization of Hamilton-Jacobi-Bellman equations

Wenjia Jing (Tsinghua University, China)

Homogenization of Hamilton-Jacobi (inviscid) equations and Hamilton-Jacobi-Bellman (viscous) equations in periodic and in stationary ergodic media has received a lot of attention since the pioneer work of Lions, Papanicolaou and Varadhan. These problems arise naturally in applications such as optimal control and combustion in heterogeneous environments. In this talk I will review some key results for the stochastic homogenization of HJB equation, and discuss some recent progress we have on the stochastic homogenization of HJ equation with a vanishing non-local term modeling jump-diffusion. Our approach is by an adaption of the method developed by Kosygina, Rezakhanlou and Varadhan (CPAM 2006). This work is joint with Dr. Qi Zhang from BIMSA.

On KPZ problem and statistics of stochastic flows

Konstantin Khanin (University of Toronto, Canada)

Calderón-Zygmund estimate for fully nonlinear equations

Shuhei Kitano (Waseda University, Japan)

The Calderón-Zygmund estimate states that the $L^p$$L^p$ norm of solutions to elliptic equations can be controlled by in terms of the $L^p$$L^p$ norm for the source terms. For linear equations it is well-known that this estimate holds for any exponents $p>1$$p>1$ but not $p=1$$p=1$. In contrast, our main result shows the estimate holds with $p=1$$p=1$ for a certain class of fully nonlinear equations.

Numerical analysis and methods for potential mean-field games

Kang Liu (Institut de Mathématiques de Bourgogne, France)

In this presentation, we address the numerical resolution of potential mean-field games (MFGs). (1) For non-degenerate MFGs, we employ a Theta-scheme for discretization, which preserves the variational structure of the continuous system. Combined with the Frank-Wolfe (FW) algorithm, this approach yields a mesh-independent convergence rate for solving the original problem. (2) We also consider first-order MFGs from a Lagrangian perspective, leading to a mean field optimization (MFO) problem. We analyze the stability of the MFO with respect to the initial distribution, which allows us to quantify the error introduced when the system is discretized using a many-particle approximation. Once again, the FW algorithm is used to solve the resulting many-particle system and provides robust approximation of the solution of first-order MFGs.

The Dirichlet problem of the homogeneous $k$$k$-Hessian equation in a punctured domain

Xinan Ma (University of Science and Technology of China, China)

we consider the Dirichlet problem for the homogeneous $k$$k$-Hessian equation with prescribed asymptotic behavior at $0\in \mathrm{\Omega }$$0\in\Omega$ where $\mathrm{\Omega }$$\Omega$ is a $(k-1)$$(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$$0$ of the solution is zero if $k>\frac{n}{2}$$k>\frac{n}{2}$, it is $\log |x|+O(1)$$\log|x|+O(1)$ if $k=\frac{n}{2}$$k=\frac{n}{2}$ and $-|x{|}^{\frac{2k-n}{n}}+O(1)$$-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$$k<\frac{n}{2}$.

To solve this problem, we consider the Dirichlet problem of the approximating $k$$k$-Hessian equation in $\mathrm{\Omega }\setminus \overline{B_r(0)}$$\Omega\setminus \overline{B_r(0)}$ with $r$$r$ small. We firstly construct the subsolution of the approximating $k$$k$-Hessian equation. Then we derive the pointwise $C^2$$C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$$0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$$(k-1)$-convex starshaped closed hypersurface in ${\mathbb{R}}^n$$\mathbb R^n$ with $\frac{n}{2}\le k<n$$\frac{n}{2}\le k<n$. This is the joint work with Zhenghuan GAO and Dekai ZHANG.

Viscosity solutions for the Newtonian $n$$n$-body problem

Ezequiel Maderna (Universidad de la República, Uraguy)

The lack of global classical solutions to the Hamilton-Jacobi equation in the $n$$n$-body problem seems to be a consequence of the well-known complexity of its dynamics. However, at positive energy levels, the Busemann functions associated with hyperbolic motions turn out to be well-defined. They are viscosity solutions, and from them it is possible to define the Green bundles of hyperbolic motions. These are invariant Lagrangian bundles, and in the case of bi-hyperbolic orbits, that is, hyperbolic both in the future and in the past, there are two bundles, and their transversality corresponds to the local solution of a scattering problem. We will present our most recent results on this topic (collaborations with Andrea Venturelli and Renato Iturriaga).

Metric-measure spacetimes: a nonsmooth approach to Einstein's theory of gravity

Robert McCann (University of Toronto, Canada)

While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually breakdown. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while and Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In a setting which relaxes local compactness, we describe a differential calculus for monotone curves and functions, and applications including a notion of infinitesmal Minkowskianity (that distinguishes Lorentz from Lorentz-Finsler norms), and a $p$$p$-d'Alembert comparison theorem which allows us to prove a Lorentzian splitting theorem for Lorentzian manifolds with limited regularity $g_{ij}\in C^1$$g_{ij} \in C^1$. Based on joint works with Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Argam Ohanyan, Felix Rott and Clemens Saemann.

Nash equilibria, Mather measures and ergodic Mean-field games

Cristian Mendico (Institut de Mathématiques de Bourgogne, France)

In this presentation, we will analyze the various domains in which the ergodic mean field game (MFG) system arises. Specifically, we will explore how weak KAM theory can be used to study this system and derive results regarding long-time behavior or the approximation of Nash equilibria. Finally, we will introduce a quasi-stationary system—a model in which, at each moment, agents optimise their expected future cost under the assumption that their environment remains static.

Convergence/divergence phenomena in the vanishing discount limit of Hamilton-Jacobi equations

Panrui Ni (Shanghai Mathematical Center, China)

We study the asymptotic behavior of solutions of an equation of the form

on a closed Riemannian manifold $M$$M$, where $G\in C(T^{\ast }M\times \mathbb{R})$$G\in C(T^*M\times\mathbb{R})$ is convex and superlinear in the gradient variable, is globally Lipschitz but not monotone in the last argument, and $c_0$$c_0$ is the critical constant associated with the Hamiltonian $H:=G(\cdot ,\cdot ,0)$$H:=G(\cdot,\cdot,0)$. By assuming that ${\partial }_{u}G(\cdot ,\cdot ,0)$$\partial_u G(\cdot,\cdot,0)$ satisfies a positivity condition of integral type on the Mather set of $H$$H$, we prove that any equi-bounded family of solutions of $\text{(*)}$$\eqref{abs_Ni}$ uniformly converges to a distinguished critical solution $u_0$$u_0$ as $\lambda \to 0^{+}$$\lambda \to 0^+$. We furthermore show that any other possible family of solutions uniformly diverges to $+\mathrm{\infty }$$+\infty$ or $-\mathrm{\infty }$$-\infty$. We then look into the linear case $G(x,p,u):=a(x)u+H(x,p)$$G(x,p,u):=a(x)u + H(x,p)$ and prove that the family $(u_{\lambda }{)}_{\lambda \in (0,{\lambda }_0)}$$(u_\lambda)_{\lambda \in (0,\lambda_0)}$ of maximal solutions to $\text{(*)}$$\eqref{abs_Ni}$ is well defined and equi-bounded for ${\lambda }_0>0$$\lambda_0>0$ small enough. When $a$$a$ changes sign and enjoys a stronger localized positivity assumption, we show that equation $\text{(*)}$$\eqref{abs_Ni}$ does admit other solutions too, and that they all uniformly diverge to $-\mathrm{\infty }$$-\infty$ as $\lambda \to 0^{+}$$\lambda \to 0^+$.

This is the first time that converging and diverging families of solutions are shown to coexist in such a generality. This is a joint work with Andrea Davini, Jun Yan and Maxime Zavidovique.

Solutions and mild solutions to general time-dependent first-order MFG systems with non-separable Hamiltonians

Antonio Siconolfi (Università di Roma, La Sapienza, Italy)

We consider a Hamiltonian $H:{\mathbb{T}}^N\times \mathbb{P}({\mathbb{T}}^N)\times {\mathbb{R}}^N\to \mathbb{R}$$H:\mathbb{T}^N \times \mathbb P(\mathbb T^N) \times \mathbb{R}^N \to \mathbb{R}$, where ${\mathbb{T}}^N$$\mathbb T^N$, $\mathbb{P}({\mathbb{T}}^N)$$\mathbb{P}(\mathbb T^N)$ stand for the $N$$N$-dimensional torus and the space of Borel probability measures on it, respectively, and assume continuity in all arguments plus convexity and superlinearity in the momentum variable.

In this setting, we find through a Lagrangian approach mild/variational solutions of the corresponding time-dependent first-order MFG system, by suitably using Kakutani–Glicksberg–Fan Theorem.

Further assuming the Hamiltonian to be differentiable in the moment variable, we prove that the mild solutions of above are actually solutions of the system in the usual sense, with continuity equation driven by a vector field depending on the solution $u$$u$ of the HJ equation, but with the advantage of being defined without requiring any additional differentiability condition on $u$$u$.

Homogenization of the Hamilton–Jacobi equation on networks

Alfonso Sorrentino (Università di Roma, Tor Vergata, Italy)

In recent years, there has been growing interest in the study of Hamilton-Jacobi equations on networks and related problems. In this talk, I will present a homogenization result for a family of time-dependent Hamilton-Jacobi equations posed on a periodic network and rescaled by a vanishing parameter. A suitable notion of periodicity, tailored to the network setting, will be introduced. As the rescaling parameter tends to zero, we obtain a limiting Hamilton-Jacobi equation defined on a Euclidean space. Notably, the dimension of this effective space is determined by the topological complexity of the network and is independent of the dimension of the ambient space in which the network is embedded. This talk is based on joint work with Marco Pozza and Antonio Siconolfi.

Singular Dynamics for Discrete Weak K.A.M. Solutions of Exact Twist Maps

Xifeng Su (Beijing Normal University)

For any exact twist map $f$$f$ and any cohomology class $c\in \mathbb{R}$$c\in\mathbb{R}$, let $u_c$$u_c$ be any associated discrete weak K.A.M. solution, and we introduce an inherent Lipschitz dynamics ${\mathrm{\Sigma }}_{+}$$\Sigma_+$ given by the discrete forward Lax-Oleinik semigroup. We investigate several properties of ${\mathrm{\Sigma }}_{+}$$\Sigma_+$ and show that the non-differentiable points of $u_c$$u_c$ are globally propagated and forward invariant by ${\mathrm{\Sigma }}_{+}$$\Sigma_+$. In particular, such propagating dynamics possesses the same rotation number ${\alpha }^{\prime }(c)$$\alpha'(c)$ as the associated Aubry-Mather set at $c$$c$.  As applications, we provide via ${\mathrm{\Sigma }}_{+}$$\Sigma_+$ a discrete analogue of Bernard's regularization theorem in 2007 and a detailed exposition of Arnaud's observation in 2011. Furthermore, we construct and analyze the corresponding dynamics on the full pseudo-graphs of discrete weak K.A.M. solutions.  This is a joint work with Jianxing Du.

Classification of Discrete Weak KAM Solutions on Linearly Repetitive Quasi-Periodic Sets

Philippe Thieullen (Université de Bordeaux, France)

A discrete weak KAM solution is a potential function that highlights the ground state configurations at zero temperature of an infinite chain of atoms interacting with a periodic or quasi-periodic substrate. It is well known that weak KAM solutions exist for periodic substrates as in the Frenkel–Kontorova model. Weak solutions may not exist in the almost periodic setting as in the theory of stationary ergodic Hamilton– Jacobi equations (where they are called correctors). For linearly repetitive quasi-periodic substrates, we show that equivariant interactions that fulfill a twist condition and a nondegenerate property always admit sublinear weak KAM solutions.We moreover classify all possible types of weak KAM solutions and calibrated configurations according to an intrinsic prefered order. The notion of prefered order is new even in the classical periodic case.

Periodic homogenization of first and second order Hamilton--Jacobi equations: recent progresses

Hung V. Tran (University of Wisconsin Madison, US)

I will give a quick introduction to the periodic homogenization theory of Hamilton--Jacobi equations and related problems. I will then describe some recent results on optimal convergence rates for both first and second order cases. Afterwards, I will explain the key ideas in the proof of the first order convex PDE.

Long-time behavior of the contact Hamilton-Jacobi semi-flow

Kaizhi Wang (Shanghai Jiao Tong University, China)

As we all know, the long-time behavior of the Hamilton-Jacobi semi-flow on the space of continuous functions on a closed manifold, where the Hamiltonian satisfies Tonelli conditions, is simple. In this talk I will introduce several recent results on the long-time behavior of the contact Hamilton-Jacobi semi-flow, where the situation is quite different from the classical case. This talk is based on joint works with Jun Yan, Yuqi Ruan and Kai Zhao.

On the dynamics of the dissipative twist map

Lin Wang (Beijing Institute of Technology, China)

For two-parameter families of dissipative twist maps, we study the dynamics of invariant graphs and the threshold for their existence and breakdown. This talk is based on some work joint with Qi Li and Alfonso Sorrentino.

Existence and nonexistence of effective burning velocity under the curvature G-equation model

Yifeng Yu (University of California Irvine, US)

G-equation is a well known level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when the curvature effect is considered:

In this talk, I will show the existence of effective burning velocity under the above curvature G-equation model when $V$$V$ is a two dimensional cellular flow, which can be extended to more general two dimensional incompressible periodic flows. Our proof combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation based on the two dimensional structures. In three dimensions, the effective burning velocity will cease to exist even for simple periodic shear flows when the flow intensity surpasses a bifurcation value. The existence result is based on joint work with Hongwei Gao, Ziang Long and Jack Xin, while the non-existence result is from collaboration with Hiroyoshi Mitake, Connor Mooney, Hung Tran, and Jack Xin.

Integrability of conformally symplectic Hamiltonian flows

Maxime Zavidovique (Sorbonne Université, France)

The goal of this talk is to introduce some notions of integrability for conformally symplectic flows associated to Tonelli Hamiltonians. We are guided by the famous result of Burago and Ivanov, that Riemannien metrics without conjugate points are flat on the torus. We prove that our conformally symplectic flows, when they don’t have conjugate points, are what we call Hopf integrable. This is joint work with Marie-Claude Arnaud and Xifeng Su.

Convergence rate of homogenization for quasi-periodic Hamilton-Jacobi equations

Jianlu Zhang (Chinese Academy, China)

In 1987, Lions firstly proposed the homogenization for Hamilton-Jacobi equations, which revealed the significance of effective Hamiltonian in controlling the large time behavior of solutions. The quantitative estimate of such a homogenization was studied in recent years, which mainly answers the convergence rate for compact case. In this talk, we will introduce a novel quasi-periodic approach, which reveals the relation between the smoothness of effective Hamiltonian and the convergence rate.

Stochastic PDE control: progresses and open problems

Xu Zhang (Sichuan University, China)

In this talk, I will present some progresses and open problems on control theory for stochastic partial differential equations. I will explain the new phenomena and difficulties in the study of controllability and optimal control problems for these systems. In particular, I will show by some examples that both the formulation of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts.