London Mathematical Society Durham Symposium

Homogenisation in Disordered Media

2018-08-19 to 2018-08-25

Abstracts of Talks

Yuri Bakhtin: *Ergodic theory of the stochastic Burgers equation*

The stochastic Burgers equation is one of the basic evolutionary SPDEs related to fluid dynamics and KPZ, among other things. The ergodic properties of the system in the compact space case were understood in 2000's. With my coauthors, Eric Cator, Kostya Khanin, Liying Li, I have been studying the noncompact case. The one force - one solution principle has been proved for positive and zero viscosity. The analysis is based on long-term properties of action minimizers and polymer measures. The latest addition to the program is the convergence of infinite volume polymer measures to Lagrangian one-sided minimizers in the limit of vanishing viscosity (or, temperature) which results in the convergence of the associated global solutions and invariant measures.

Yves Capdeboscq: *On Microstructures with sign changing Jacobians*

In this talk we will discuss recent results regarding micro-structures generating sign changing Jacobians. This question has appeared in the study of a family of parameter reconstruction problem called "hybrid inverse problems", but was previously also investigated, with a different purpose in mind, in the context of homogenization theory. We show that using quantitative corrector results, one can construct new families of sign changing micro-structures. A work in progress regarding the connection with regularity theory will also be mentioned.

Annalisa Cesaroni: *Fattening phenomena for nonlocal curvature flows*

I will discuss geometric evolutions by nonlocal curvature flow with a general kernel. In particular I will consider regularity of the evolution in terms of the fattening phenomenon, and I will present some examples of fattening vs nonfattening singularities. This is based on a joint work with M. Novaga, S. Dipierro, E. Valdinoci.

Paul Dario: *Homogenization on percolation cluster*

The theory of stochastic homogenization is generally developped in the setting of uniformly elliptic environment. In this talk we show how to adapt this theory in a setting where the environment is not uniformly elliptic: the case of the infinite cluster in supercritical percolation. We will present a renormalization structure of the infinite cluster which is the main technical argument to adapt the theory from the uniformly elliptic setting to the percolation cluster as well as the main results obtained by this approach: quantitative homogenization, large scale regularity, optimal bounds on the corrector. This is joint work with S. Armstrong.

Andrea Davini: *Stochastic homogenization of viscous and non-viscous HJ equations with non-convex Hamiltonians*

I will present a homogenization result in dimension 1 for viscous and non-viscous Hamilton-Jacobi equations with non-convex Hamiltonians in random media. The result is new in the viscous case. This talk is based on a joint work with Elena Kosygina.

Federica Dragoni : *Stochastic Homogenisation in Carnot groups*

Ben Fehrman: *A Liouville theorem for stationary and ergodic ensembles of parabolic systems*

We will discuss a first-order Liouville theorem for random ensembles of uniformly parabolic systems under the qualitative assumptions of stationarity and ergodicity. In particular, the analysis will yield a quantitative homogenization estimate in terms of the sublinear growth of a canonical extended corrector. The sublinearity of the corrector provides the starting point for a Campanato iteration. We will use this iteration to establish, almost surely, an intrinsic large-scale Hölder estimate for caloric functions, from which the Liouville theorem follows.

William Feldman: *Shapes of local minimizers for a model capillary energy in periodic media*

I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy. Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion.

Martin Hairer: *Random directed mean curvature flow*

Inwon Kim: *Large time behavior for mean curvature flow with periodic forcing*

In this talk we will study large time behavior of mean curvature flow, where the forcing term is positive, periodic and Lipschitz. We will introduce a notion of maximal and minimal speed for the flow, and show that homogenization holds when they coincide, and linear growth of long fingers occur when they do not. In the graph setting we will show the existence of maximal and minimal traveling waves. This is joint work with Hongwei Gao.

Claude Le Bris: *Elliptic homogenization with defects*

We will present some recent mathematical contributions related to nonperiodic homogenization problems. The difficulty stems from the fact that the medium is not assumed periodic, but has a structure with a set of embedded localized defects, or more generally a structure that, although not periodic, enjoys nice geometrical features. The purpose is then to construct a theoretical setting providing an efficient and accurate approximation of the solution. This is a series of joint works with X. Blanc, PL Lions, and M. Josien.

Xue-Mei Li: *Stochastic Averaging*

We study perturbations of stochastic dynamics. This corresponds to singular perturbations of parabolic differential equations of second order. We will explain some natural models and the related geometric metods.

Jessica Lin: *Stochastic Homogenization for Reaction-Diffusion Equations*

I will present several results concerning stochastic homogenization for reaction-diffusion equations. We consider heterogeneous reaction-diffusion equations with stationary and ergodic nonlinear reaction terms. Under certain hypotheses on the environment, we show that the typical large-time, large-scale behavior of solutions is governed by a deterministic front propagation which can be identified by a Hamilton-Jacobi equation. This talk is based on joint work with Andrej Zlatos.

Jeta Molla: *Multiscale Modelling of Li-batteries*

We introduce a simple and basic lithium battery formulation that captures the essential transport and reaction processes. We state explicit boundary conditions for a voltage- and a current-driven battery. The latter is generally the case of interest in applications. After identifying a characteristic representative volume element for specific composite cathodes materials, we derive effective macroscopic reactive transport equations. The herewith systematically derived homogenized transport formulation contains the same fundamental feature, i.e., interfacial reactions becoming properly weighted bulk terms, as motivated by Newman and collaborators using volume-averaging. This is joint work with M. Schmuck and S. Atalay.

Stefan Neukamm: *Quantitative homogenization in nonlinear elasticity*

We consider a nonlinear elastic composite with a periodic microstructure described by the nonconvex energy functional $\int_\Omega W(\tfrac x\epsilon,\nabla u(x))-f(x)\cdot u(x)\,dx.$ It is well-known that under suitable growth conditions the energy $\Gamma$-converges to a homogenized functional a homogenized energy density $W_{hom}$. One of the main problems in homogenization of nonlinear elasticity is that long-wavelength buckling prevents the possibility of homogenization by averaging over a single period cell, and thus $W_{hom}$ is in general given by an infinite-cell formula. Under appropriate assumptions on $W$ (frame indifference, minimality at identity, non-degeneracy) and on the microstructure (e.g., possibly touching smooth inclusions), we show that in a neighbourhood of rotations $W_{hom}$ is characterized by a single-cell homogenization formula. In particular, we prove that correctors are available—a property that we exploit to derive a quantitative two-scale expansion and uniform Lipschitz estimates for minimizers. This is joint work with Mathias Schäffner (TU Dresden).

Matteo Novaga: *Optimal configurations for the Heitmann-Radin energy*

We consider low energy configurations for the Heitmann-Radin sticky discs functional, in the limit of diverging number of discs. More precisely, we renormalize the Heitmann-Radin potential by subtracting the minimal energy per particle, i.e., the so called kissing number. For configurations whose energy scales like the perimeter, we prove a compactness result which shows the emergence of polycrystalline structures: The empirical measure converges to a set of finite perimeter, while a microscopic variable, representing the orientation of the underlying lattice, converges to a locally constant function.

Marcel Ortgiese: *Branching random walks in random environment*

We will consider a branching random walk on a lattice, where the branching rates are locally given by a random potential. Since there are various sources of randomness, the central question is how the system compares to the corresponding averaged versions. When averaging only over branching and migration, the expected number of particles solves a heat equation with a random potential known as the parabolic Anderson model. Over the last decade there has been considerable progress in understanding the latter, driven by the observation that the system exhibits intermittency. In our work we concentrate on the effect of averaging over branching/migration and try to understand if the particle system is close to the solution of the heat equation. It turns out that the answer will depend essentially on the extreme value behaviour of the underlying potential.

Felix Otto: *Multipole expansion in random media*

In a homogeneous medium, the far-eld generated by a local source is well-described by the multipole expansion, the coecients of which are given by the moments of the charge distribution. In case of a random medium that homogenizes, this is not covered by standard homogenization theory, since the source lives on a scale comparable to the correlation length. However, the constant-coecient situation survives, intrinsically interpreted, to some degree: In three space dimensions, the analogy holds up to quadrupoles. This insight allows for identifying the best artificial boundary conditions for a finite computational domain. This is joint work with P. Bella & A. Giunti, and with JF. Liu, based on work with J. Fischer.

Greg Pavliotis: *Long time behaviour and phase transitions for the McKean-Vlasov equation*

We study the long time behaviour and the number and structure of stationary solutions for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction. Our main objectives are the study of convergence to a stationary state and the construction of the bifurcation diagram for the stationary problem. The application of our work to the study of models for opinion formation is also discussed.

Christophe Prange: *Boundary layers in periodic homogenization: some recent developments*

This talk is concerned with quantitative periodic homogenization in domains with boundaries. The quantitative analysis near boundaries leads to the study of boundary layers correctors, which have in general a nonperiodic structure. The interaction between the boundary and the microstructure creates geometric resonances, making the study of the asymptotics or continuity properties particularly challenging. The talk is based on work with S. Armstrong, T. Kuusi and J.-C. Mourrat, as well as work by Z. Shen and J. Zhuge.

Mariya Ptashnyk: *Coupling between mechanics and chemistry: Multiscale modelling and analysis*

Many biological tissues must be structured in such a way as to be able to adapt to two extreme biomechanical scenarios: they have to be strong enough to resist high pressure and mechanical forces and yet also be flexible to allow large expansion and growth. A part of nature's solution to this intriguing problem is the complex microstructures of biological tissues together with interconnected microscopic (cellular) processes. To analyse the interplay between the mechanics, microscopic structure and the chemistry in a biological tissue we derive microscopic models for plant biomechanics and assume that the elastic properties of cell walls depend on the chemical processes, whereas chemical reactions depend on mechanical stresses within the cell walls. The microscopic models constitute strongly coupled systems of reaction-diffusion-convection equations for chemical processes and equations of elasticity or poroelasticity for elastic deformations and growth. To analyse the macroscopic behaviour of plant tissues, the macroscopic models for plant biomechanics are derived using homogenization techniques. In the multiscale analysis we distinguish between periodic and random distribution of cells in a plant tissue. Numerical solutions for macroscopic models demonstrate heterogeneity in the cell wall displacement due to interactions between mechanical stresses, microstructure, and chemical processes.

Claudia Raithel: *A Large-Scale Regularity Theory for Random Elliptic Operators on the Half-Space*

We derive an almost-sure first-order Liouville principle for linear elliptic equations on the half-space with homogeneous Dirichlet boundary data, assuming an ensemble of coefficient fields defined on the whole-space that is stationarity and satisfies a quantified ergodicity condition. We obtain this Liouville principle as a corollary of a $C^{1,\alpha}$-decay of an associated excess, which we prove via a Campanato iteration that takes as input a sublinear extended corrector that is adapted to the boundary data. While the proof of the excess decay follows the previous work of Gloria, Neukamm, and Otto, the main difficulty in our work is the construction of the half-space corrector and the associated flux corrector. This is a joint work with Julian Fischer.

Jean-Michel Roquejoffre: *Front propagation in the presence of a line with large diffusion: a property of the level sets.*

The situation is the following: a line, having a strong diffusion on its own, exchanges mass with a reactive medium, in our case a two-dimensional strip. A front propagates both on the line and in the strip, and one wishes to describe its shape. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by transportation networks. Numerical simulations, due to A.-C. Coulon, reveal an a priori surprising property: the solution is not monotone in the direction orthogonal to the strip. The goal of this talk is to explain this feature, in the particular case of a reaction modelled by a Dirac measure carried by the set where the solution reaches its minimum value. The main part of the work consists in providing an asymptotic expansion of the solution in the neighbourhood of the point where the free boundary hits the line. Its conclusions confirm the numerical simulations, at least qualitatively. Joint work with L. Caffarelli.

Ben Seeger: *Scaling limits and homogenization for some stochastic Hamilton-Jacobi equations*

I present some results concerning homogenization of Hamilton-Jacobi equations that are perturbed by multiplicative or additive noise in time. In the multiplicative case, under certain conditions on the spatial environment and on the mixing nature of the noise, the large-time, large-space behavior of the equation is governed by the solution of a spatially homogenous, stochastic Hamilton-Jacobi equation. On the other hand, a Hamilton-Jacobi equation with an additive noise perturbation that has weak strength but fast spatial oscillations may exhibit various types of behavior. In particular, convergence to infinity, vanishing of the noise, or stochastic homogenization can occur, the latter being observed in the scaling critical case.

Charles Smart : *A free boundary problem with facets*

Valery Smyshlyaev: *Two-scale homogenisation of micro-resonant PDEs (periodic and some stochastic)*

Macroscopic properties of composite materials containing "micro-resonances" can be very different from those of conventional materials. Mathematically this leads to studying homogenisation of problems with a "critically" scaled high contrast, where the resulting two-scale asymptotic behaviour appears to display a number of interesting effects. We will review some background, as well as some more recent developments and applications. One is two-scale analysis of general "partially-degenerating" periodic PDE problems [1], where strong two-scale resolvent convergence of high-contrast elliptic operators appears to hold under a rather generic decomposition assumption. This implies in particular (two-scale) convergence of semigroups with applications to a wide class of micro-resonant dynamic problems, both parabolic and hyperbolic. We also discuss situations where the micro-resonances display certain randomness. In simplest cases, the resulting two-scale limit behaviour appears to be rather explicit and the macroscopic equations display a wave localization i.e. a kind of trapping by the micro-resonances due to their randomness. [1] I.V. Kamotski, V.P. Smyshlyaev, Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients, to appear in Applicable Analysis (2018), published online 27 February 2018. Available also as https://arxiv.org/pdf/1309.4579v2.pdf

Florian Theil: *Are atomistic equilibrium distributions ordered?*

It is an open problem whether ordered crystalline structures can be explained as equilibrium configurations of atomistic energies at low temperatures. We demonstrate that in two dimensions configurations exhibit orientiational order if dislocations are tightly bound. This is a consequence of rigidity estimates. Therefore the central question is whether the neutrality constraint is actually an obstruction to disorder. First results in this direction are shown for the Ariza-Ortiz' model for discrete crystal elasticity and discrete dislocations in crystals.

Hung Tran: *Min-max formulas for nonconvex effective Hamiltonians and application to stochastic homogenization*

We introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex Hamiltonians, which are applicable in both periodic and general stationary ergodic settings. Secondly, we analytically and numerically investigate other related interesting phenomena, such as quasi-convexification and breakdown of symmetry. Joint work with Qian and Yu.

Yifeng Yu: *Optimal rate of convergence in periodic homogenization of Hamilton-Jacobi equations*

In this talk, I will present some recent progress in obtaining the optimal rate of convergence $O(\epsilon)$ in periodic homogenization of Hamilton-Jacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(\epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system. This allows us to employ powerful tools from the Aubry-Mather theory and the weak KAM theory. It is a joint work with Hiroyashi Mitake and Hung V. Tran.

Ofer Zeitouni : *Homogenization for the multiplicative stochastic heat equation in dimension $d\geq 3$*

I will prevent several convergence results for solutions of the stochastic heat equation with multiplicative coloured noise, after rescaling and recentering. Based on joint works with Yu Gu, Lenya Ryzhik and Alex Dunlap.