Effective properties of interface evolution in a random environment

Lead Research Organisation: Cardiff University
Department Name: Sch of Mathematics

Abstract

This is mathematical research connecting the analysis of partial differential equations (PDEs) and (applied) probability.

The main goal of this project is to derive macroscopic evolution laws for interfaces in heterogeneous, random environments, described on a small scale by nonlinear PDEs with random coefficients. In particular, we want to go beyond classical homogenisation in order to treat systems where a long-range collective behaviour emerges. A key aspect of this project is to establish a new link between analysis and probability, benefitting both fields, by working on problems motivated by applications.

It is motivated by the following situation:

With an interface is associated a scalar quantity called its energy (think e.g. of its area) which it tries to decrease, i.e., it performs a gradient flow. This energy is perturbed through obstacles or impurities on a very small scale, and driven by some large-scale force. The impurities are random, i.e., we have information only on the probability of finding certain impurities in a certain place, not on their precise nature and location.

We are interested in the effective velocity and other qualitative properties of the interface on a large scale, much larger than the scale on which the perturbations vary. On that scale, the perturbations should average out, but the questions is:

What is the effective evolution law on a large scale, and what are the qualitative properties of the interface, e.g. on which scales does it look rough due to all the random heterogeneities? How does all this depend on the law of the impurities?

This is important because we are interested in the reaction of a system on the scale of our everyday life to an input on that scale. E.g. we would like to know how a piece of metal changes shape in a car crash, we are not interested in the position of each single atom, and we wouldn't be able to compute those anyway. But most realistic materials have some random structure on a fine scale.

Planned Impact

Our main beneficiaries are

1. The PDRAs through training at the interface between analysis (partial differential equations) and probability and through the networking opportunities our workshops and network of international collaborators provide.
2. The mathematical community through novel results on effective evolution laws and powerful techniques. We will reach out to these beneficiaries through publications both in leading peer-reviewed journals and open-access depositories, moreover we will disseminate our results on international conferences and host two workshops.
3. On a medium-to-long timescale, the applied sciences (physics, engineering, life sciences) will benefit as many important processes (spreading of species, crack propagation, dislocation motion) can be modelled by partial differential equations in a random environment.
A precise understanding of the multi-scale nature of the solutions to such equations is important for creating fast codes for numerical computations. We will reach out to these communities by presenting e.g. at interdisciplinary or engineering conferences and by inviting specialists from those sciences to our final workshop.
4. On a larger timescale, our research will through its impact on applied sciences also benefit the industrial sector, e.g. a better understanding of the connection between microscopic material properties and macroscopic behaviour in engineering will ultimately lead to more powerful tools for designing novel materials.
5. Finally we will reach out to the general public by public lectures, for example explaining and illustrating the connection of scale-bridging and randomness to everyday life.

Publications

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Cesaroni A (2016) Homogenization of a Mean Field Game System in the Small Noise Limit in SIAM Journal on Mathematical Analysis

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Dirr N (2019) Some new results on Lipschitz regularization for parabolic equations in Journal of Evolution Equations

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Embacher P (2018) Computing diffusivities from particle models out of equilibrium. in Proceedings. Mathematical, physical, and engineering sciences

 
Description 1. We have obtained results how stochastic system are connected to macroscopic quantities in two important examples" Nonlinear diffusion equations, which are macroscopical gradient flows, and the Ising model
2. We have obtained new upscaling tools, i.e mathematical methods for rigorously deriving large-scale equations from small-scale equations.
Exploitation Route May be of interest for the physics/engineering communiy
Sectors Other

 
Description Collaboration with group of Pierre Cardaliaguet 
Organisation Paris Dauphine University
Country France 
Sector Academic/University 
PI Contribution Expertise limits for in interacting particle systems, interface evolution models
Collaborator Contribution Expertise in stochastic homogenisation for Hamilton-Jacobi equations
Impact Funding application to Royal Society
Start Year 2016