**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.

The log log scale for compressible advection equations

This talk aims at presenting some recent quantitative estimates, obtained with D. Bresch, for transport or advection equations with rough velocity fields in an appropriate Sobolev space but in the presence of potentially strong compression effects. When the divergence of the velocity field is bounded (or at least when the flow is nearly-incompressible), the classical theory of renormalized solutions provides a log regularity on the flow; but when the flow is compressible with potentially large or vanishing values of the density, no estimates were known. However, using a new duality approach, we have been able to identify a log log scale of regularity that is propagated by the equation. Our final goal is to use those estimates to obtain new results on complex systems where the transport equation is coupled to other PDE’s: A driving example is the compressible Navier-Stokes system.

From slow diffusion to a hard height constraint: a singular limit of Keller-Segel

For a range of physical and biological processes—from dynamics of granular media to biological swarming—the evolution of a large number of interacting agents is modeled according to the competing effects of pairwise attraction and (possibly degenerate) diffusion. In the slow diffusion limit, the degenerate diffusion becomes a hard height constraint on the density of the population, as arises in models of pedestrian crown motion. Motivated by these applications, we bring together new results on the Wasserstein gradient flow of nonconvex energies with the theory of free boundaries to study the slow diffusion limit of the Keller-Segel equation. Our analysis demonstrates the utility of Wasserstein gradient flow as a tool to construct and approximate solutions, alongside the strength of viscosity solution theory in examining their precise dynamics. We then apply these results to develop numerical evidence for open conjectures in geometric shape optimization. This is based on joint works with Inwon Kim, Ihsan Topaloglu, and Yao Yao.

Stochastic homogenization of a nonlinear PDE modelling hysteresis

Motivated by the modelization of hysteresis observed in some materials, we consider a time-dependent nonlinear equation with coefficients that vary randomly on a small lengthscale. At the fine scale, the model exhibits a hysteretic behavior which persists under asymptotically slow loading. Using stochastic two-scale convergence as introduced by A. Bourgeat, A. Mikelic and S. Wright, ideas developed by A. Mielke and the fact that the fine scale model is convex, we identify its homogenized limit. This coarse model here writes as a corrector equation and a coarse equation, which are coupled one with each other. We also show that, at the coarse scale, the model again exhibits hysteretic behavior which persists under slow loading. Joint work with Thomas Hudson (Warwick) and Tony Lelievre (ENPC).

Determining modes for the Navier-Stokes equation

We will review classical results on determining modes for fluid equations and present a slightly different approach where we start with a time-dependent determining wavenumber defined for each individual trajectory and then study its dependence on the force. While in some cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution. For the 3D Navier-Stokes equation, we obtain a uniform bound on the time average of this wavenumber, which we estimate in terms of the Kolmogorov dissipation number and Grashof constant.

Nekhoroshev stability for the Schrodinger flow in a quasi-periodic potential

The random heat equation in dimensions three and higher

We consider the heat equation with a multiplicative Gaussian potential in dimensions three and higher. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with an additive white noise, with an effective diffusion and an effective variance. This is a joint work with Alex Dunlap, Yu Gu and Ofer Zeitouni.

For questions, contact Chris Henderson at: henderson