Global Existence and Computer-Assisted Proofs of Singularities in Incompressible Fluids

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The dynamics of free surfaces moving with incompressible fluids occurs in many problems in engineering and science. For short times, the behaviour of solutions is understood in many cases. However, the theory of existence (or not) of singularities, and long term behavior of solutions, is far from well-developed. An example of singularities are the breaking of waves, the formation of a tornado or the splash of a drop.This project takes two directions: on the one hand it addresses the fundamental question of whether there exists breakdown of smooth solutions in finite time, paying particular attention to the type of breakdown and to the quantity that blows up and on the other the existence of global solutions that exist for all time. Two equations are considered: the generalized surface quasi-geostrophic (gSQG) equation- both in the smooth and the patch case -, a family of models which interpolates between the surface quasi-geostrophic (SQG) equation and the Euler vorticity equation; and the two dimensional free boundary Euler equations. In order to carry out the project, a combination of techniques among an interdisciplinary tool set is required. These include, but are not restricted to, mathematical analysis, high-performance numerical computing, and rigorous computations leading to computer-assisted proofs.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.