Analysis & Applications Seminar

SEMINARIO DOBLE -- On the least action principle applied to Rayleigh-Taylor subsolutions // Homogenization in General Relativity

Speaker:  Bjorn Gebhard, Universidad Autónoma de Madrid // André Guerra, ETH Zurich ()
Date:  Friday, 23 September 2022 - 10:30 // 11:30
Place:  Aula 520, Módulo 17, Departamento de Matemáticas, UAM


10:30. "On the least action principle applied to Rayleigh-Taylor subsolutions", Bjorn Gebhard (Universidad Autónoma de Madrid)

Abstract: We consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is called the Rayleigh-Taylor instability. In the talk we will see how infinitely many admissible solutions to the Euler equations reflecting a turbulent mixing of the two fluids can be constructed. The solutions are induced by subsolutions, which can be seen as horizontal averages, via convex integration. Since there are also infinitely many subsolutions, a selection problem for the averaged motion arises. Motivated by the fact that the Euler equations themselves can be derived by the least action principle, we will discuss the consequences of imposing the least action principle on the level of subsolutions.
The talk covers material of joint works with Jóef Kolumbán, Lászlo Székelyhidi, Jonas Hirsch.


11:30. "Homogenization in General Relativity", André Guerra (ETH Zurich)

Given their highly nonlinear nature, Einstein’s equations are not closed under weak convergence and hence sequences of weakly-convergent solutions may generate a non-trivial energy momentum tensor in the limit. In 1989 Burnett conjectured that, for a sequence of vacuum solutions which oscillates with high frequency, this limit is characterized by the Einstein-massless Vlasov model: in particular, starting from vacuum, matter is generated through homogenization. In this talk we will present a proof of this conjecture, under appropriate gauge and symmetry assumptions. Based on joint work with Rita Teixeira da Costa (Princeton University).