Geometric problems in PDEs with applications to fluid mechanics
Principal investigator: Alberto Enciso
Project reference: 633152
Start Date: March 2015
End Date: February 2020
Abstract: There are many high-profile problems in PDEs that ultimately boil down to assertions of a strongly geometric ortopological nature. One feature that makes these problems both very difficult and extremely appealing is that there is not a standard set of techniques that one can routinely resort to in order to attack them. Indeed, the very nature of these questions makes them strongly interdisciplinary, so successful approaches require finely tailored combinations of ideas and techniques coming from different branches of mathematics (analysis, geometry and topology), often interspersed with some physical intuition. In this project I aim to go significantly beyond the state of the art in a wideclass of geometric questions in PDEs, with an emphasis on problems in fluid mechanics and encompassing long-standing questions that can be traced back to leading analysts and geometers such as Arnold, De Giorgi and Yau. The project is divided in three interrelated blocks, respectively devoted to the study of Beltrami fields in steady incompressible fluids, to geometrice volution problems and to global approximation theorems. Key to the proposal is a versatile new approach to a number of geometric problems in PDEs that I have pioneered and applied in several seemingly unrelated contexts. The power of this technique is laid bareby my recent proofs of a well-known conjecture on knotted vortex lines in topological fluid mechanics that was popularized by Arnold and Moffatt in the 1960s and of a long-standing conjecture on the existence of thin vortex tubes in steady solutions to the Euler equation that dates back to Lord Kelvin in 1875.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme grant agreement No 633152.