**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 Chris Henderson.

Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample

In order to build a solution of a critical Hamilton-Jacobi (HJ) equation, Lions, Papanicolaou and Varadhan (1986) consider the associated discounted HJ equation, and prove that the solution converges uniformly along any subsequence to a solution of the critical equation, as the discount factor vanishes. Recently, Davini, Fathi, Iturriaga and Zavidovique (2016) proved that for convex Hamiltonian, the limit is unique. The question remained open for nonconvex Hamiltonians. In this work, we build a 1-dimensional continuous and coercive (nonconvex) Hamiltonian such that this limit is not unique. The construction is based on a discrete-time repeated game example, called stochastic game. No game theory prerequisite is needed understand the talk, so please come!

Singularity Formation in Incompressible Fluids

We will discuss some recent results on finite-time and infinite-time singularity formation for strong solutions to the incompressible Euler equations and related fluid models. Our basic approach is to use scale-invariance to derive simpler lower dimensional equations where singularity formation can be established in a relatively straightforward fashion. Based on joint works with I. Jeong.

Stochastic homogenization and generic regularity

We consider stochastic homogenization of elliptic equations in divergence form, in the general ergodic case. Following the strategy of Avellaneda an Lin of ``borrowing'' the regularity theory from the homogenized operator, and inspired by Armstrong and Smart, we show that there is a random almost-surely finite radius from which onwards $C^{1,\alpha}$-Schauder theory and $H^{1,p}$-Calderon-Zygmund theory kicks in.

This is joint work with A. Gloria and S. Neukamm.

Characterizing fluctuations in stochastic homogenization

We consider stochastic homogenization of elliptic equations in divergence form and are interested in characterizing the leading-order fluctuations of macroscopic observables $\int g\cdot\nabla u$ of a general solution $\nabla\cdot(a\nabla u+f)=0$. It turns out that these can be related in a pathwise fashion to those of the corrector $\phi$, provided the two-scale expansion $(1+\phi_i\partial_i)\bar u$ is inserted into the homogenization commutator $a\nabla u-\bar a\nabla u$. This allows to characterize fluctuations in terms of a four-tensor $\bar Q$, which in turn can be extracted from the Representative Volume Element method at no further cost than extracting the homogenized coefficient $\bar a$ itself.

This is joint work with M. Duerinckx and A. Gloria.

Effective multipoles in random media

We show that, suitably re-interpreted, the multipole expansion of a solution $\nabla\cdot(a\nabla u+f)=0$ in the $d$-dimensional space and its relation to the moments of the compactly supported r.h.s. $f$ also hold in a heterogeneous medium $a$. However, in the random as opposed to the periodic case, the order of the expansion is limited to $\frac{d}{2}$, through the growth of higher-order correctors. This effective multipole theory is build on a canonical duality between (quotients of) spaces of $a$-harmonic functions that grow at a certain order and those of $a$-harmonic functions that decay at a certain order. The (higher-order) two-scale expansion provides an isomorphism between these spaces and the classical spaces related to the homogenized coefficient $\bar a$ that preserves the duality form.

This is joint work with P. Bella and A. Giunti.

Homogenization theory in the presence of defects: an update on the current results

The context is homogenization theory for linear elliptic equations in the presence of microscopic defects. The talk will overview the general abstract problem, and the two specific cases of a local defect and a twin interface. It is successively shown that the corrector exists, in the appropriate functional space, that it enjoys suitable properties, that it allows to actually obtain an accurate approximation of the oscillatory solution and to make precise the rates of this approximation. Some works in progress will also be mentioned.

This is a series of joint works with Xavier Blanc (Université Paris Diderot) and Pierre-Louis Lions (Collège de France), and also Marc Josien (Ecole des Ponts).

Smoothing results for the Landau equation

This talk is concerned with the spatially inhomogeneous Landau equation from plasma physics. After giving an overview of prior work, we will discuss two recent regularity results. First, a smoothing theorem (obtained jointly with C. Henderson) that says weak solutions are $C^\infty$ provided the mass, energy, and entropy are bounded above, and the mass is bounded below. Second, a positive lower bound on the mass density (obtained jointly with C. Henderson and A. Tarfulea) that allows one to remove the lower bound on the mass from the smoothing criteria. The main tools are, respectively, the iteration of local Schauder estimates, and the analysis of a stochastic process associated to the equation via a formula of Feynmann-Kac type. We will also briefly discuss the implications of these results for the existence theory of the Landau equation.

May 2

May 9

June 6 (*Special Time*)

For questions, contact Chris Henderson at: henderson