Concentration and quantitative regularity in homogenization and hydrodynamics

Christophe Prange (1)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

This habilitation thesis is about a selection of my works concerned with the study of regularity for Partial Differential Equations, with a focus on equations stemming from fluid dynamics. In a broad sense, regularity theory is the study of the local behavior of solutions. In our view the main objectives are: (i) to identify a range of scales where there is a certain self-similar behavior, (ii) to find basic objects i.e. building blocks, that represent well the solutions at these scales, (iii) to prove scaling laws for excess quantities, i.e. local error estimates, at these scales. Our analysis is motivated by physics: concentration effects in composite materials, fluids slipping over rough surfaces, generation of turbulence near boundaries in fluids. Our aim is to contribute to analyzing such phenomena from the perspective of regularity theory. There are two main parts in this thesis. The first part is concerned with large-scale regularity and quantitative homogenization of three-dimensional stationary Navier-Stokes and elliptic equations. With Higaki (former postdoctoral researcher, now at Kobe University) and Zhuge (The University of Chicago), we prove large-scale regularity results in bumpy regions possibly as rough as fractals. With Armstrong (NYU), Kuusi (University of Helsinki) and Mourrat (ENS de Lyon), we obtain near-optimal error estimates for the homogenization of boundary layer correctors. The second part is concerned with the three-dimensional non stationary Navier-Stokes equations. With Maekawa (Kyoto University) and Miura (Tokyo Institute of Technology), we find new pressure estimates that enable us to control the strong nonlocality in the half-space. With Albritton (IAS, Princeton) and Barker (University of Bath), we investigate norm and geometric concentration near potential singularities. We also establish a connection between concentration and quantitative regularity in the critical case that leads to a slight breaking of the criticality barrier.

Mots clés: Fluid dynamics; Navier-Stokes; Homogenization; Boundary layers; Quantitative regularity; Concentration; Singularities

Mots clés: Mécanique des fluides; Navier-Stokes; Homogénéisation; Couches limites; Régularité quantitative; Concentration; Singularités

Domaines: Equations aux dérivées partielles [math.AP]

Christophe Prange. Concentration and quantitative regularity in homogenization and hydrodynamics. Analysis of PDEs [math.AP]. CY Cergy Paris Université, 2022. ⟨tel-03814044⟩