Qualitative Properties of Solutions to Fluids Equations

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This project addresses the solutions of partial differential equations modeling fluids. The principal goal is understanding the evolution of an incompressible fluid. In particular, the PI will study physical problems involving fluids with a free interface. This includes the motion of water waves, with or without surface tension, and a fluid-structure interaction problem. The PI will introduce techniques which shall advance our knowledge of fluids and promote further applications in science and engineering. Graduate students will be involved in all aspects of the project.For the fluid structure interaction system, the research will seek a priori estimates leading to the local and long-time existence of solutions and the construction of solutions satisfying these properties. The PI will also study the qualitative properties of solutions, such as the persistence of analytic and Sobolev regularity, along with asymptotic behavior. Furthermore, the research seeks to answer similar questions for the solutions of the Euler equations with a free interface, focusing on the local existence and uniqueness of solutions. The PI will work on properties of solutions of other important models in connection with fluid dynamics, such as the 2D and 3D Boussinesq equations, active scalar equations, and the Prandtl equations.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.