### 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.

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

TBC

## Abstract

TBC

#### Maciej Capinski (AGH, Poland)

TBC

## Abstract

TBC

#### 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)

TBC

## Abstract

TBC

#### Alex Haro (University of Barcelona, Spain)

Sun-Jupiter-Saturn system may exist: a verified computation of quasiperiodic solutions for the planar three body problem

## Abstract

In this talk, we present evidence of the stability of a simplified model of our Solar System, a flat (Newtonian) Sun-Jupiter-Saturn system with realistic data, that is, with masses of the Sun and the planets, their semi-axes, eccentricities and precessions of the planets close to the real ones. The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasi-periodic motion, that lies in an invariant torus, with the appropriate frequencies. To do so, first KAM schemes to compute translated tori are used to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is invariant tori. Second, KAM schemes for invariant tori are used to refine the solution of the invariant equations giving the invariant torus. Last, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem. This is a joint work with Jordi-Lluís Figueras (Uppsala University).

#### Jonathan Jaquette (New Jersey Institute of Technology, USA)

Perspectives on computing infinite dimensional (stable) manifolds in PDEs with validated numerics

## Abstract

Dynamical systems offer a rich set of tools for analyzing the long-term behavior of solutions to evolutionary equations, and many finite dimensional techniques may be extended to analyze partial differential equations. However to develop a validated numerical approximation of stable manifolds, one is met with several difficulties: Proof techniques that work for finite dimensional problems or for discrete systems simply do not work for PDEs; and the stable manifold is an infinite dimensional object, so any sort of high-order approximation of it quickly encounters the curse of dimensionality. A way forward may be found with the Lyapunov-Perron method. While impractical to compute with, it allows one to set up a contraction mapping on the space of graphs over the stable eigenspace. A no-frills version of the theorem can be nicely obtained for a linear approximation of the stable manifold, but in order to take advantage of a higher order approximation one must leverage a detailed analysis of the semi-group operator in a transformed coordinate system. I will present how this approach may be applied to yield computer-assisted-proofs of stable manifolds in the Swift-Hohenberg equation, and discuss future directions and challenges.

#### Mioara Joldes (LAAS-CNRS, France)

Optimization-Aided Construction of Multivariate Chebyshev Polynomials

## Abstract

This talk is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one looks for best approximants to specific monomials by polynomials of lower degree relative to the uniform norm. Exploiting the Moment-SOS hierarchy, we devise a semidefinite-programming-based procedure to compute such best approximants, as well as associated signatures. Applying this procedure in three variables leads to the values of best approximation errors for all monomials up to degree six on the euclidean