The importance of Luis Caffarelli's work in the study of fluids
Maria J. Esteban
TL;DR
This survey highlights Luis Caffarelli's transformative contributions to fluid mechanics, focusing on partial regularity for the 3D Navier–Stokes equations, global regularity for the critical surface quasi-geostrophic equations, and the regularity of free boundaries and capillary interfaces. It emphasizes methodological advances such as $\varepsilon$-regularity schemes, monotonicity formulas, De Giorgi-type arguments, and the weighted Caffarelli–Kohn–Nirenberg inequalities, which have become foundational across both local and nonlocal PDEs. The discussion also shows how non-fluid PDE techniques, including fractional diffusion and homogenization, were innovatively adapted to fluid problems, enriching the analytical toolkit for complex interfaces and free-boundary phenomena. Collectively, these results underpin a large portion of modern regularity theory and continue to influence analyses of jets, Stefan problems, and capillary drops in fluids.
Abstract
In this paper we describe the work of Luis Caffarelli in the area of fluid mechanics and related topics. Not only has his work on fluid mechanics been very influential, but many of his contributions that do not directly relate to fluid mechanics, such as his important results on the fractional Laplacian or the regularity of solutions to linear parabolic equations with oscillating coefficients, have been used in the study of fluids in many important ways. Thus, any review of his work has to include his contributions to the general (partial) regularity theory of solutions of Navier-Stokes equations and other studies related to fluid motion.