### RESEARCH TALKS

#### Gianni Arioli (Polytechnic University of Milan, Italy)

Epochs of irregularity of Leray-Hopf solutions to 3D Navier-Stokes equations

## Abstract

We study global Leray-Hopf solutions to Cauchy problems for the 3D Navier-Stokes equations in a cube under Navier boundary conditions. Since the pioneering work by Jean Leray it is known that finite energy solutions exist for any initial data, but it is not known yet whether their enstrophy may blow up in finite time in the so-called epochs of irregularity. We present computer assisted results which bring strong evidence to the fact that the enstrophy blow-up may indeed occur in finite time.

#### Nicolas Brisebarre (Ecole Normale Superieure de Lyon, France)

Series Expansions in Classical Orthogonal Polynomials for Solutions of Linear Differential Equations

## Abstract

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. Thanks to a framework due to Ore, we interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms for computing a suitable recurrence relation. (This is joint work with Alexandre Benoit and Bruno Salvy).

#### Florent Brehard (Ecole Polytechnique Lille, France)

Validated Numerics for Multiple Root-Finding of Univariate and Bivariate Polynomials

## Abstract

A posteriori validation methods based on fixed-point theorems are used extensively in the field of validated numerics. In most cases, the problem is locally invertible around the solution and tight error bounds can be obtained by using a Newton(-like) operator. For singular (and thus degenerate) problems, however, one typically needs to solve overdetermined systems rigorously, which admit no solution under generic, arbitrarily small perturbations. Therefore, even properly specifying what a validation algorithm should validate in such cases is a nontrivial question. In this talk, we will illustrate this challenge for degenerate polynomial root-finding in two cases.

#### Maciej Capinski (AGH, Poland)

Oscillatory collision approach in the Earth-Moon restricted three body problem

## Abstract

We consider the Earth-Moon planar circular restricted three body problem and present a proof of the existence orbits, which approach arbitrarily close to one of the primary masses, and at the same time after each approach they move away from the mass to a prescribed distance. In other words the orbits oscillate between being arbitrarily close to collision and away from it. The proof is based on the fact that the family of Lyapunov orbits and the associated transversal intersections of their stable/unstable manifolds can be continued to a collision with a primary.

#### Maximilian Engel (University of Amsterdam, Netherlands)

Detecting random bifurcations via rigorous enclosures of large deviations rate functions

## Abstract

We provide a description of transitions from uniform to nonuniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of moment Lyapunov exponents which are principal eigenvalues of the generator of the tilted (Feynman-Kac) semigroup. Using a computer assisted proof, we demonstrate how to determine these eigenvalues and investigate the rate function which is the Legendre-Fenichel transform of the moment Lyapunov function. We apply our results to two case studies: the pitchfork bifurcation and a two-dimensional toy model, also considering the transition to a positive asymptotic Lyapunov exponent. This is joint work with Alexandra Blessing, Alex Blumenthal and Maxime Breden.

#### Jordi-Lluis Figueras (Uppsala, Sweden)

KAM algorithms for invariant tori

## Abstract

In this talk I wil