For a bounded domain, we consider the L^\infty-functional involving a nonnegative Hamilton function. Under the continuous Dirichlet boundary condition and some assumptions of Hamiltonian H, the uniqueness of absolute minimizers for Hamiltonian H is established. This extendes the uniqueness theorem to a larger class of Hamiltonian $H(x,p)$ with $x$-dependence. As a corollary, we confirm an open question on the uniqueness of absolute minimizers posed by Jensen-Wang-Yu. Our proofs rely on geometric structure of the action function induced by Hamiltonian H(x,p), and the identification of the absolute subminimality with convexity of the associated Hamilton-Jacobi flow.
In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) x (the undisturbed water depth) > 1.363.... In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simpler approximate models to gain insights.
I will begin by Whitham's shallow water equation and the modulational instability index for small amplitude and periodic traveling waves, the effects of surface tension and vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is partly based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), and Ashish Pandey (Illinois).
In the first part of this talk I will do a brief introduction to the recent theory of Mean Field Games (MFG) initiated by J.-M. Lasry and P.-L. Lions. The main objective of the MFG theory is the study of the limit behavior of Nash equilibria for symmetric differential games with a very large number of “small” players. In its simplest form, as the number of players tends to infinity, limits of Nash equilibria can be characterized in terms of the solution of a coupled system of a Hamilton-Jacobi and Fokker-Planck (or continuity) equations. The first equation describes the evolution of the value function of a typical agent, while the second one characterizes the evolution of the agents’ density. In the second part, I will introduce a variational MFG model, where we impose a density constraint. From the modeling point of view, imposing this new constraint means that we are aiming to avoid congestion among the agents. We will see that a weak solution of the system contains an extra term, an additional price imposed on the saturated zones. I will show that this price corresponds to the pressure field from the models of incompressible Euler equations à la Brenier. If time permits, I will discuss the regularity properties of the pressure variable, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model. The talk is based on a joint work with P. Cardaliaguet (Paris Dauphine) and F. Santambrogio (Paris-Sud, Orsay).
We consider this problem expressed in position-velocity potential holomorphic coordinates. We will explain the set up of the problem(s) and try to present the advantages of choosing such a framework. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. The talk will try to be self contained.