Boundaries, Oscillations, layeRs in Differential Systems – BORDS

This grant proposal is motivated by two things. Firstly, many physical or biological models are multiscale and heterogeneous. Phenomena with multiple scales are found everywhere from the physics of superconductivity, material science or biology to geophysics. A common feature of these systems is that they involve strong heterogeneities: roughness (sea floor), porosity (petroleum reservoirs), or oscillations in their properties (composite materials). Secondly, a lot of instabilities and singular behaviors (spike effects, creation of vorticity in fluids, onset of turbulence) arise near boundaries and are created by a subtle interaction between boundary conditions, nonlinearities and oscillations.

While oscillations in coefficients of Partial Differential Equations (PDEs) are by now much better understood, much less is known about the effect of singularities in the domain or in the boundary data. Our project focuses on the study of the boundary behavior of solutions. The combination of oscillations in the domain or in the boundary condition forces concentration effects and intricate dynamics near boundaries.

The primary goals of the grant proposal are: (i) to investigate the interplay between oscillations, geometry or nonlinearities which may lead to complex dynamics of the solution, (ii) to achieve a better understanding of the effect of boundary conditions on the solutions of PDEs or on the spectrum of operators by analyzing singular cases, (iii) to deal with non smooth or singular boundaries (Lipschitz domains, corners), (iv) to bring together experts in homogenization, spectral theory and fluid mechanics to investigate boundary induced instabilities in fluids.

Our main directions of research will be:
[Singular boundary data] Our group will study differential systems with singular boundary data: highly oscillating boundary data and singular mixed boundary conditions. Our goal is to analyze the effect of strong oscillations, or large parameters in the boundary data in various geometric settings, possibly in domains having low regularity or corners. The singularities in the boundary data force concentration phenomena near the boundary and lead to the study of boundary layer problems. One leitmotiv of the team is to achieve error estimates and formulas making our results tractable for numerical analysis.
[Asymptotics in highly oscillating domains] Systems with rough or highly disordered boundaries are ubiquitous in nature and industry. Applications involve scales ranging from microfluidics to large scale oceanic currents, and very different types of fluids (salt water, liquid Earth core, glaciers, polymeric fluids) or matter (living cells, aquifers). Our goals are twofold: tackle delicate mathematical issues in the analysis of boundary or interface layers, and use our theoretical progress to address real-world models from geophysics (ocean currents, water waves, flow of glaciers over a rough bedrock or a deformable bed of sediments) and biology (electroporation, interfaces between tissues).
[Instabilities in fluids] Many instabilities develop close to boundaries, due to the complex combination of boundary conditions and nonlinearities constitutive of the fluid: high Reynolds instabilities, elastic instabilities in viscoelastic fluids, roughness-induced instabilities. These instabilities may result in the propagation of high frequency waves, in energy transfers to small scales and eventually in the transition to high Reynolds or elastic turbulence. Our team aims at getting a better grasp on such phenomena, which is crucial to many industries notably the aeronautic industry. Our goal is to investigate such problems in three different contexts: (i) high Reynolds instabilities for viscoelastic flows, (ii) roughness-induced or -reduced instabilities, (iii) instabilities in glaciology. This part will benefit from progress on the first two directions and mobilize the skills of all team members.