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Can every physical system simulate any Turing machine? This is a classical problem that is intimately connected with the undecidability of certain physical phenomena. Concerning fluid flows, Moore [C. Moore, Nonlinearity 4, 199 (1991)] asked if hydrodynamics is capable of performing computations. More recently, Tao launched a program based on the Turing completeness of the Euler equations to address the blow-up problem in the Navier¿Stokes equations. In this direction, the undecidability of some physical systems has been studied in recent years, from the quantum gap problem to quantum-field theories. To the best of our knowledge, the existence of undecidable particle paths of three-dimensional fluid flows has remained an elusive open problem since Moore¿s works in the early 1990s. In this article, we construct a Turing complete stationary Euler flow on a Riemannian S3 and speculate on its implications concerning Tao¿s approach to the blow-up problem in the Navier¿Stokes equations., Robert Cardona was supported by the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Program for Units of Excellence in R&D (MDM-2014-0445) via an FPI grant. R.C. and E.M. are partially supported by Grants MTM2015-69135-P/FEDER, the Spanish Ministry of Science and Innovation PID2019-103849GB-I00/AEI/10.13039/501100011033, and Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR932. E.M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. D.P.-S. is supported by MICINN Grant MTM PID2019-106715GB-C21 and MCIU Grant Europa Excelencia EUR2019-103821. F.P. is supported by MICINN/FEDER Grants MTM2016-79400-P and PID2019-108936GB-C21. This work was partially supported by ICMAT–Severo Ochoa Grant CEX2019-000904-S.

Vlasov equations can be formally derived from N-body dynamics in the mean-field limit. In some suitable singular limits, they may themselves converge to fluid dynamics equations. Motivated by this heuristic, we introduce natural scalings under which the incompressible Euler equations can be rigorously derived from N-body dynamics with repulsive Coulomb interaction. Our analysis is based on the modulated energy methods of Brenier and Serfaty., Minor typos corrected

Lagrange algorithm (CLA) for the incompressible Euler equations. Accurate simulations of periodic flow confined in a cylinder with walls are pursued in sight of a recently reported finite time singularity (blow-up) in this geometry. The algorithm exploits the time-analyticity of the particle trajectories of the velocity field. The time-Taylor coefficients of the trajectories fulfill recursion relations and are obtained by solving Helmholtz-Hodge decomposition problems. A Chebyshev-Fourier pseudo-spectral approach to the appearing numerical problems in the underlying cylindrical coordinate system is studied while avoiding unstable spectral differentiation. The appearing Poisson problems are directly solved for the second derivatives of their solutions. For this, a sophisticated Poisson solver, based on modified Bessel functions, is developed and employed. Time-stepping techniques, fixed and adaptive, are proposed and numerically investigated. Furthermore, several methods for the scattered interpolations of the flow fields, known on the particle trajectories after a time step insertion, back onto the fixed grid are reviewed and compared. The presented implementation of the CLA aims for accuracy and is found to be stable and fast also due to the absence of a CFL condition. The latter allows larger time steps and thus reduces tremendously the total number of iterations. Thorough tests on stationary and swirl-free flows are executed and discussed, before a number of simulations of the blow-up candidate from before are inspected. The implementation of the CLA that is presented in this thesis proves to be an excellent tool for fast and accurate incompressible Euler computations in the given geometry.; À l'aide du nouvel algorithme de Cauchy-Lagrange (CLA) ce travail étudie un écoulement d'Euler incompressible, axisymétrique et périodique, confiné dans un domaine cylindrique, borné par une paroi dans la direction radiale. Des simulations très précises sont effectuées en vue de reconsidérer le résultat récent d'une possible singularité en temps fini (blow-up) pour un flot particulier dans cette géométrie. L'algorithme exploite l'analyticité temporelle des trajectoires des particules du champ de vitesse. Les coefficients de Taylor en temps des trajectoires satisfont des relations de récursion et sont obtenus en résolvant des problèmes de décomposition de Helmholtz-Hodge. Une approche pseudo-spectrale du type Chebyshev-Fourier est appliquée aux problèmes numériques dans un système aux coordonnées cylindriques tout en évitant une différenciation spectrale instable. Des méthodes pour le choix d'un pas de temps, fixe ou adaptatif, sont proposées et étudiées numériquement. Après l'insertion d'un pas de temps, les champs d'écoulement sont connus sur les trajectoires des particules dispersés dans une tranche du cylindre. Plusieurs méthodes d'interpolation sur une grille déformée sont considérées et comparées afin de trouver la meilleur façon de rétablir le flot sur la grille fixe qui donne la nouvelle condition initiale pour le mécanisme de récursion. L'implémentation du CLA présentée se révèle être précise, stable et rapide. L'absence de condition CFL rend le CLA plus performant que les algorithmes traditionnels. En effet, cette propriété permet de prendre des pas de temps plus large, ce qui réduit le nombre total d'itérations. Des tests approfondis sur les écoulements stationnaires et d'autres, non-stationnaires, sans vitesse angulaire sont effectués. Puis, des simulations avec des conditions initiales susceptibles de développer une singularité en temps fini, sont effectuées et analysées. Cette première mise en œuvre du CLA pour un domaine avec une paroi s'avère être un excellent outil pour résoudre les équations d'Euler incompressibles d'une manière rapide et précise dans une géométrie donnée.

In this article we study the limit when $\alpha \to 0$ of solutions to the $\alpha$-Euler system in the half-plane, with no-slip boundary conditions. We establish the existence of subsequences conv...

We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich-Bures metric space, which is a matricial analogue of the Wasserstein and Hellinger-Kantorovich metric spaces. We establish some topological, metric and geometric properties of this space, which includes the existence of the optimal transportation path.

This papers deals with the large time behavior of solutions of the incompressible Euler equations in dimension 2. We consider a self-similar configuration of point vortices which grows like the squ...

International audience; We approximate the regular solutions of the incompressible Euler equations by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold's interpretation of the solution of the Euler equations for incompressible and inviscid fluids as geodesics in the space of measure-preserving diffeomorphisms, and an extrinsic approximation of the equations of geodesics due to Brenier. Using recently developed semi-discrete optimal transport solvers, this approach yields a numerical scheme which is able to handle problems of realistic size in 2D. Our purpose in this article is to establish the convergence of this scheme towards regular solutions of the incompressible Euler equations, and to provide numerical experiments on a few simple test cases in 2D.

In the vast literature on tsunami research, few articles have been devoted to energy issues. A theoretical investigation on the energy of waves generated by bottom motion is performed here. We start with the full incompressible Euler equations in the presence of a free surface and derive both dispersive and non-dispersive shallow-water equations with an energy equation. It is shown that dispersive effects only appear at higher order in the energy budget. Then we solve the Cauchy-Poisson problem of tsunami generation for the linearized water wave equations. Exchanges between potential and kinetic energies are clearly revealed., Comment: 20 pages, 12 figures. Accepted to Proceedings of the Royal Society A. Other authors papers and supporting material can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy.