Understanding and classifying stationary solutions for the 2D Euler equations is a major problem that has implications in other fields (e.g. convex integration), other equations (SQG, Navier-Stokes) and would improve the understanding of turbulence.
An important achievement of this project that has gone beyond the state of the art is the set of results concerning rigidity and flexibility of solutions to the 2D Euler and more general active scalar equations. The first part of this result (rigidity) has been published in Duke Math. Journal, one of the leading journals in mathematics and will lead to further developments. The second part (flexibility) is under consideration at a top journal.
We have proved the existence of finite time imploding singularities for 3D compressible Euler for all values of the adiabatic exponent, and for Navier-Stokes, The key idea to go from Euler to Navier-Stokes is that for certain self-similar solutions, under certain conditions in the scaling exponent, the viscosity is a lower order term and can be treated as an error. We believe this has implications beyond the compressible case, in particular in the incompressible one, as well as many other equations (Cordoba-Cordoba-Fontelos model, SQG....).
We have developed a new framework, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations. The solution in addition corresponds to an asymptotic self-similar profile for the 3-dimensional Euler equations in the presence of a cylindrical boundary. In particular, the solution represents a precise description of the Luo-Hou blow-up scenario [G. Luo, T. Hou, Proc. Natl. Acad. Sci. 111(36): 12968-12973, 2014] for 3-dimensional Euler. To the best of the authors' knowledge, the solution is the first truly multi-dimensional smooth backwards self-similar profile found for an equation from fluid mechanics. The new numerical framework is shown to be both robust and readily adaptable to other equations.
We expect the combination of the techniques mentioned in the previous 2 paragraphs to yield very fruitful outcomes in the future, solving longstanding open problems.
On a different topic, we have started to develop tools to tackle hard open questions in spectral geometry (such as the hotspots conjecture from the 1970’s or the Polya-Szego from the 1950’s). We have written two papers. This is more foundational work than results-oriented and sets up a path for the resolution of the aforementioned problems.
