Regularity and Asymptotic Behavior in Fluid Dynamics

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Partial differential equations are essential for studying natural phenomena such as fluid flow, heat transfer, and wave propagation. They help model the dynamics of the atmosphere, ocean, stars, and other physical phenomena. The aim of this project is to study partial differential equations that model fluid dynamics and other complex systems that involve fluids, allowing for improved understanding of fluid properties and control. Mathematical models describing fluid interactions with solid and elastic bodies are also of interest, with applications to engineering and other scientific fields. This project also provides training and research opportunities for graduate students. This project addresses the existence, regularity, and qualitative properties of solutions to partial differential equations and systems modeling fluids interacting with elastic bodies. The project considers fluid-structure interaction models that involve fluids interacting with elastic solids, for both compressible or incompressible fluids, and in viscous and inviscid settings. The primary focus is on establishing global existence, uniqueness, and asymptotic behavior of the solutions. Additionally, the project addresses the existence and qualitative behaviors of solutions to fluid-structure interaction models involving plates and elastic membranes. Furthermore, the project studies the inviscid limit problem and other singular limits, either for the Navier-Stokes equations in bounded domains or in more involved physical settings. The project shall also address existence and regularity issues related to fluid models with evolving boundaries. One important example is the Euler system with a free interface, both with and without surface tension. Finally, the project will consider questions concerning the Boussinesq system, including those relating to the persistence of regularity and the long-horizon behavior of solutions.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.