The seminar meets regularly on Wednesdays at 4pm in Eckhart 202. We also have special seminars during other days. To subscribe or unsubscribe the email list, you may either go to Camp/PDE email list or contact Will Feldman.
The least-action problem for geodesic distance on the `manifold' of fluid-blob shapes exhibits instability due to microdroplet formation. This reflects a striking connection between Arnold's least-action principle for incompressible Euler flows and geodesic paths for Wasserstein distance. A connection with fluid mixture models via a variant of Brenier's relaxed least-action principle for generalized Euler flows will be outlined also. This is joint work with Jian-Guo Liu and Dejan Slepcev.
The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, this population-dynamics model accounts for the spatial spreading and growth of a species. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards I will describe the new model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation by using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.
I will describe a "geometric" approach to proving the strong maximum principle and its application to obtain results for solutions of degenerate elliptic PDEs like those of M. Bardi and F. Da Lio, and (with modification) for the norm of the gradient of a solution to the infinity Laplacian as done by Y. Yu. Then, by constructing appropriate approximations to the solutions of uniformly elliptic fully nonlinear PDEs, I will describe how to extend the strong maximum principle to the difference of pairs of solutions for such a PDE.
I shall present some recent results about some singular perturbations of Dirac operator and its connection with the boundedness of the Cauchy operator and Calderon’s projector operator. Also I will sketch the proof of an isoperimetric type inequality.
In a joint work with A. Porretta (Roma 2), we study the long time behavior of the master equation in Mean Field Games (MFG) theory. This equation is a kind of transport equation in the space of measures. We show that its solution converges to a (weak) solution of an ergodic master equation as time tends to infinity. The non standard difficulty consists in estimating a uniform regularity of the solution with respect to the measure variable. In a similar way, we analyze the convergence, as the discount factor tends to 0, of the discounted MFG system.