**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.

Congested crowd motion and quasi-static evolution

In this talk we investigate the relationship between a quasi-static evolution and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. We will also discuss preliminary results on general initial data.

Optimal control of conditioned processes and Mean Field Games

October 12

Interfaces, junctions and stratification (I)

Interfaces, junctions and stratification (II)

Stochastic Homogenization for Reaction-Diffusion Equations

We study heterogeneous reaction-diffusion equations in stationary ergodic random media ignition nonlinearities. Under suitable hypotheses on the environment, we prove the existence of deterministic asymptotic speeds of propagation for solutions with both compactly supported and front-like initial data. We subsequently obtain a general stochastic homogenization result which shows that, in the large-scale-large-time limit, the behavior of typical solutions in such environments is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. This talk is based on joint work with Andrej Zlatos.

Lecture 1: Taming infinities

Monday, October 25, 2016, 4:30pm–5:30pm, Ryerson 251

Abstract: Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

Lectures 2 and 3: The BPHZ theorem for stochastic PDEs

Tuesday, October 26, 2016, 4:30pm–5:30pm, Eckhart 202

Wednesday, October 27, 2016, 4pm–5pm, Eckhart 202

Abstract: The Bogoliubov-Parasiuk-Hepp-Zimmermann theorem is a cornerstone of perturbative quantum field theory: it provides a consistent way of "renormalising" the diverging integrals appearing there to turn them into bona fide distributions. Although the original article by Bogoliubov and Parasiuk goes back to the late 50s, it took about four decades for it to be fully understood. In the first lecture, we will formulate the BPHZ theorem as a purely analytic question and show how its solution arises very naturally from purely algebraic considerations. In the second lecture, we will show how a very similar structure arises in the context of singular stochastic PDEs and we will present some very recent progress on its understanding, both from the algebraic and the analytical point of view.

Min-max representation for operators satisfying the global comparison property

A mapping \(F\) between spaces of real valued functions is said to have the "global comparison property" (GCP) if \(u\leq v\) everywhere with \(u=v\) at some point \(x\) means that \(F(u)\leq F(v)\) at this point \(x\). A classical result of Courrège says that a continuous linear map from \(C^2(\mathbb{R}^d)\) to \(C^0(\mathbb{R}^d)\) has the GCP if and only if it is a sum of jump and drift-diffusion operators. In work with Russell Schwab, we characterize nonlinear maps having the GCP as those given by a min-max of linear operators having the GCP. This result in particular provides a representation formula for the Dirichlet-to-Neumann map of elliptic equations, and for the interface velocity in free boundary problems such as Hele-Shaw.

Nonlinear echoes and Landau damping with insufficient regularity

In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space for the 1D periodic case. This is demonstrated by constructing arbitrarily small solutions with a sequence of nonlinear oscillations, known as plasma echoes, which damp at a rate arbitrarily slow compared to the linearized Vlasov equations. Some connections with hydrodynamic stability problems will be discussed if time permits.

Homogenization theory in the presence of defects: Correctors do correct

The context is homogenization theory for linear elliptic equations in the presence of local microscopic defects. The purpose of the talk is to establish that the correctors obtained in the work Blanc/LeBris/Lions (Comm. in PDE, 2015) indeed provide an accurate approximation of the oscillatory solution, in the expected norms and at the microscopic scale, and to make precise the rates of this approximation, depending upon the integrability and regularity of the measure defining the microscopic defects. The argument is based on works by Avellaneda and Lin, and Kenig, Lin and Shen.

The work is joint work with Xavier Blanc (Université Paris Diderot) and Marc Josien (Ecole des Ponts).

November 30

Uncertainty quantification for multiscale kinetic equations with uncertain coefficients

In this talk we will study the generalized polynomial chaos-stochastic Galerkin (gPC-SG) approach to kinetic equations with uncertain coefficients/inputs, and multiple time or space scales, and show that they can be made asymptotic-preserving, in the sense that the gPC-SG scheme preserves various asymptotic limits in the discrete space. This allows the implemention of the gPC methods for these problems without numerically resolving (spatially, temporally or by gPC modes) the small scales. Rigorous analysis will be provided to prove that these schemes are stochastically asymptotic preserving. Examples to be discussed include the linear neutron transport equation and the nonlinear Boltzmann/Landau equations.

For questions, contact Will Feldman at: feldman