Regularity and Stability Analysis of Free-Boundary Problems in Fluid Dynamics

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Applications of fluid dynamics are ubiquitous in science and engineering, ranging from biology to geology, oceanography, and aerospace. This project focuses on a class of mathematical models commonly encountered in practical fluid dynamics applications: free-boundary problems. In such systems, fluid flow is modeled by the solution to a partial differential equation formulated in a domain whose boundary is dynamic and evolves according to couplings with the fluid fields. Free-boundary problems are among the most mathematically challenging in fluid dynamics and more generally in the analysis of partial differential equations due to their notorious complexity, severe nonlocality, and implicit nonlinearity. The broad objective of this project is to develop new methods to advance understanding of this class of models. The project aims to study the long-time existence, regularity, and behavior of generic solutions, as well as the stability of special solutions. The project will also provide opportunities for involvement of graduate students in the research.The project will study three models of different nature: the Muskat problem, water waves, and the free-boundary incompressible porous medium equation. Regarding the Muskat problem for fluids with constant density in porous media, the aims are to establish global existence and uniqueness and investigate long-time behavior of large solutions for the one-phase problem. Construction of special solutions and their stability will be another focus. For water waves, the project intends to rigorously demonstrate the instability of the classic two-dimensional Stokes waves for both two-dimensional and three-dimensional perturbations with and without surface tension effects. The free-boundary incompressible porous medium equation will be investigated regarding local well-posedness for a large class of density profiles and stability analysis for steady states. All three equations are quasilinear and are either degenerate parabolic or hyperbolic or mixed. A set of tools from harmonic analysis, microlocal analysis, potential theory, spectral theory, and bifurcation theory will be combined and sharpened to tackle these challenging questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.