### EXPOSÉS

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

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

## Résumé

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

## Résumé

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

## Résumé

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

## Résumé

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

## Résumé

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

## Résumé

In this talk I will give a broad survey of different settings where we are using reliable KAM algorithms for computing invariant tori. These algorithms can be converted into proofs. This is joint work with Rafael de la Llave, Joan Gimeno, Jeremy Parker; and Chiara Caracciolo and Alex Haro.

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

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

## Résumé

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

## Résumé

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

## Résumé

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 ball, the simplex, and the cross-polytope. Inspired by numerical experiments, we obtain explicit expressions for Chebyshev polynomials in two cases unresolved before. This is joint work with M. Dressler, S. Foucart, E. de Klerk, J.-B. Lasserre and Y. Xu.

#### Xuefeng Liu (Tokyo Woman’s Christian University, Japan)

Solution Verification for the Navier-Stokes equation in 3D domains: large-scale computation and axisymmetric solutions

## Résumé

Explicit error estimation for the finite element solution to the Navier-Stokes equation enables the verification of stationary solutions in 3D domains. However, the substantial computational resources required make it challenging to study flows at high Reynolds numbers. To address this issue, we are currently developing efficient computational algorithms utilizing parallel computation on the Fugaku supercomputer. Additionally, we are exploring axisymmetric solutions in cylindrical domains, which have recently yielded interesting results concerning the potential blow-up of solutions. In this talk, I will introduce the processes and findings of these approaches.

#### Michael Plum (Karlsruhe Institute of Technology, Germany)

Non-selfadjoint spectral problems related to self-similar blowup in nonlinear wave equations

## Résumé

#### Elena Queirolo (Technical University of Munich, Germany)

DelayDE, periodic orbits and bifurcations

## Résumé

In this talk, we present some results on validation of branches of periodic orbits in DelayDE. We draw your attention towards the peculiarities of DDE problem and we apply this technique in conjunction with the validation of multi-parameter solutions. This cocktail allows us to prove the existence of a branch of Hopf bifurcations in DDEs, up until the Hopf bifurcation itself disappears.

#### Akitoshi Takayasu (University of Tsukuba, Japan)

Computer-assisted proofs for a projected boundary value problem

## Résumé

In recent work, we have focused on rigorous forward integration in time for semilinear parabolic partial differential equations (PDEs) via semigroup theory. This approach formulates a zero-finding problem on the Banach space of space-time variables to find a solution of time-evolutionary PDEs. By finding a fixed-point of the Newton-like operator for this zero-finding map, we can prove the existence and local uniqueness of the solution in the neighborhood of an approximate solution with computer-assistance. It is worth noting that the Fréchet derivative of the map at the approximate solution is characterized by the linearized operator, and the uniqueness of the solution to the linearized problem allows the explicit formulation of the inverse operator of the Fréchet derivative. In this talk, we extend this methodology by formulating a projected boundary value problem as a similar zero-finding problem. By combining with the explicit form of the inverse with the usual computer-assisted approach, we demonstrate a computer-assisted approach for boundary value problems via semigroup theory. This approach is directly applicable to the boundary value problems of semilinear parabolic PDEs, providing a framework for further applications. This is joint work with Jean-Philippe Lessard (McGill University).

#### Warwick Tucker (Monash, Australia)

The Songling system has exactly four limit cycles

## Résumé

Determining how many limit cycles a planar polynomial system of differential equations can have is a remarkably hard problem. One of the main difficulties is that the limit cycles can reside within areas of vastly different scales. This makes numerical explorations very hard to perform, requiring high precision computations, where the necessary precision is not known in advance. Using rigorous computations, we can dynamically determine the required precision, and localise all limit cycles of a given system. We prove that the Songling system of planar, quadratic polynomial differential equations has exactly four limit cycles. Furthermore, we give precise bounds for the positions of these limit cycles using rigorous computational methods based on interval arithmetic. The techniques presented here are applicable to the much wider class of real-analytic planar differential equations. This is joint work with Zbigniew Galias.

#### Thomas Wanner (George Mason University, USA)

Bifurcation point validation using extended systems

## Résumé

Parabolic partial differential equations are fundamental models for many applied phenomena. For example, pattern formation in diblock copolymers is described by the evolution of the fourth-order Ohta-Kawasaki equation, while competition in the interaction of populations can be modeled by second-order reaction-diffusion systems such as the Shigesada-Kawasaki-Teramoto model. Essential for a deeper understanding of the long-term dynamics of such problems are the sets of equilibria as described by the associated bifurcation diagrams. In this talk, we present computer-assisted proof techniques which can be used to validate and continue bifurcation points through the use of suitable extended systems. This includes not only fold points, but also pitchfork and transcritical bifurcations which are the result of group actions beyond forcing through involutions. Our results apply to both one- and two-dimensional domains, and they can also be used to treat certain bifurcation points with higher-dimensional kernels.

#### Daniel Wilczak (Jagiellonian University, Poland)

C^1 convergence for Galerkin projections of dissipative PDEs

## Résumé

I will present several results about convergence of solutions to Galerkin projections and associated variational equation. These theoretical results justify correctness of the C^1-like algorithm for a class of dissipative PDEs. The talk is based on a join paper with Piotr Zgliczyński https://arxiv.org/abs/2405.17087.

#### JF Williams (Simon Fraser University, Canada)

TBC

## Résumé

TBC

#### Piotr Zgliczynski (Jagiellonian University, Poland)

The pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE

## Résumé

We present the tools for dissipative PDEs that allow to describe the dynamics near the pitchfork bifurcation. In particular, the tools allow us to verify the existence of the heteroclinic connections born after the bifurcations. We also give conditions which allow to perform a rigorous computer-assisted proof that the established structure exists on an explicit parameter range. We apply our methods to two bifurcations from zero for the Kuramoto–Sivashinsky PDE. The obtained parameter range is big enough to allow us to continue the fixed points and the heteroclinic connections using the rigorous integration.