Computer-assisted Proofs in Fluid Mechanics and Applications

The research goal of this project is twofold: on the one hand to pursue methods and ideas developed in recent work in the search for either singularities or global existence in incompressible fluids with finite energy and on the other to transfer the techniques to solve long-standing open problems in spectral geometry and mathematical physics. A key ingredient in its success is to have accurate numerics, together with a deep understanding of the regularity theory. Therefore, the interdisciplinary nature of the project, which involves numerical computations, physics informed neural networks (PINN), computer-assisted proofs, modern methods in partial differential equations (PDE) and harmonic analysis, is an essential ingredient for its successful outcome. The recent emergence of techniques blending mathematical analysis and deep learning offers a new opportunity for developing new angles of attack and better understanding classical problems in mathematics. Undergraduate and graduate students, postdoctoral researchers, and research visitors actively participate in the work of this project. In particular, undergraduate students from underrepresented groups in the mathematical sciences are mentored and involved in research experiences integral to the project, on problems that are rich and suitable for learning and partial progress, both numerically and analytically. The project is structured in three blocks, the first one involving the incompressible Euler equations, as well as other water wave models; the second on the surface quasi-geostrophic (SQG), the generalized-SQG equations, and related models; and a third one on applications to spectral geometry and mathematical physics. 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.