Periodic Reporting for period 3 - COR-RAND (Corrector equations and random operators)
Période du rapport: 2023-09-01 au 2025-02-28
Physical models used to understand the world are often written in the form of partial differential equations, which encode nontrivial relations between the characteristic quantities of the system (such as temperature, velocity, position, external forces, time, etc.). This physics often involves several scales (the scale of observation, the scale of microscopic constituents, etc.) and one important question is to understand how to pass from one scale to the other (think of a micrometer versus a meter), and deduce reduced models (which are hopefully better suited to numerical simulations e.g.).
One prototypical feature of "microscopic models" is the inherent presence of randomness and defects. One may think of a suspension of sugar molecules in water, the bottom of a river made of a random arrangement of pebbles, or a periodic crystal with random defects (typically due to thermal fluctuations e.g.). Understanding the averaged effect of this microscopic randomness on a larger scale is the aim of this ERC project. More precisely the project addresses the interaction between randomness (which models the physical description of the system) and differential operators (which model physical laws and how the system evolves) from a mathematical perspective.
One prototypical feature of "microscopic models" is the inherent presence of randomness and defects. One may think of a suspension of sugar molecules in water, the bottom of a river made of a random arrangement of pebbles, or a periodic crystal with random defects (typically due to thermal fluctuations e.g.). Understanding the averaged effect of this microscopic randomness on a larger scale is the aim of this ERC project. More precisely the project addresses the interaction between randomness (which models the physical description of the system) and differential operators (which model physical laws and how the system evolves) from a mathematical perspective.
First, we have analyzed the long time behavior of acoustic waves in heterogeneous media. Due to the interaction of the waves by the heterogeneities, their long-time description is challenging. We have characterised the wave to leading order by the solution of an effective wave equation for which heterogeneities are averaged out. The maximum time scale for which the result holds depends on the type of heterogeneities, and is minimal in the random setting. Capitalizing on this we have then provided lower bounds for the localization length of localized eigenvectors for the acoustic operator at the bottom of the spectrum.
Second, we have studied random suspensions I a fluid flow, and more specifically random suspensions of rigid particles in a steady Stokes flow.
In particular,
- We have rigorously derived the homogenized system for neutrally-buoyant particles, which takes the form of some Stokes fluid with an effective viscosity.
- Based on the quantitative theory, we have addressed the case of sedimenting particles and successfully defined the so-called effective sedimentation speed. This has allowed us to revisit and rigorously analyse the celebrated Caflisch-Luke paradox, and rule out this paradox under the assumption of hyperuniformity/
- We have given the most general justification of the Einstein formula for the effective viscosity (and adapt the argument to the Clausius-Mossotti formula), and developed a general theory for the expansion at arbitrary order.
- Suspensions of swimming bacteria can lead to a decrease of the effective viscosity in physical experiments. We have introduced and analysed a model for such active suspensions; and rigorously justified the drastic reduction of viscosity predicted by the physics literature and observed experimentally.
Third, we have started to extend the quantitative homogenization theory available for linear elliptic equations to the nonlinear setting, and established a quantitative two-scale expansion for random monotone operators (which model nonlinear heat conduction e.g.).
Fourth, in order to connect spectral theory of random operators to PDE analysis, we have defined and studied the correlations of the landscape function in the whole space, a tool of great practical interest in the physics of Anderson localisation.
Second, we have studied random suspensions I a fluid flow, and more specifically random suspensions of rigid particles in a steady Stokes flow.
In particular,
- We have rigorously derived the homogenized system for neutrally-buoyant particles, which takes the form of some Stokes fluid with an effective viscosity.
- Based on the quantitative theory, we have addressed the case of sedimenting particles and successfully defined the so-called effective sedimentation speed. This has allowed us to revisit and rigorously analyse the celebrated Caflisch-Luke paradox, and rule out this paradox under the assumption of hyperuniformity/
- We have given the most general justification of the Einstein formula for the effective viscosity (and adapt the argument to the Clausius-Mossotti formula), and developed a general theory for the expansion at arbitrary order.
- Suspensions of swimming bacteria can lead to a decrease of the effective viscosity in physical experiments. We have introduced and analysed a model for such active suspensions; and rigorously justified the drastic reduction of viscosity predicted by the physics literature and observed experimentally.
Third, we have started to extend the quantitative homogenization theory available for linear elliptic equations to the nonlinear setting, and established a quantitative two-scale expansion for random monotone operators (which model nonlinear heat conduction e.g.).
Fourth, in order to connect spectral theory of random operators to PDE analysis, we have defined and studied the correlations of the landscape function in the whole space, a tool of great practical interest in the physics of Anderson localisation.
Concerning acoustic waves in heterogeneous media, we aim at relating quantitative homogenization to Anderson localisation at the lower spectrum -- where both theories apply. This will give a quantitative control on the localization length in the lower spectrum. More generally, we aim at getting nontrivial information on random operators using time-dependent models in random media.
The results on random suspensions in fluids now give a rather complete picture in the setting of steady Stokes flows. One last work in this direction is the justification of the celebrated Batchelor formula for the sedimentation speed.
The results on random suspensions in fluids now give a rather complete picture in the setting of steady Stokes flows. One last work in this direction is the justification of the celebrated Batchelor formula for the sedimentation speed.