Équation d'Euler-Lagrange

Résumé

L’équation d'Euler-Lagrange (en anglais, Euler–Lagrange equation ou ELE) est un résultat mathématique qui joue un rôle fondamental dans le calcul des variations. On retrouve cette équation dans de nombreux problèmes réels de minimisation de longueur d'arc, tels que le problème brachistochrone ou bien encore les problèmes géodésiques. Elle est nommée d'après Leonhard Euler et Joseph-Louis Lagrange. Notations E désignera un espace vectoriel normé, [t , t] un intervalle réel, et \mathcal G l'espace affine des fonctions x : [t , t] → E de classe C telles que x\left(t_i\right)=x_i, où x , x sont deux vecteurs fixés de E. Le vecteur dérivé d'une fonction x\in\mathcal G en un point t ∈ [t , t] est noté \dot x(t). On se donne par ailleurs une fonction \mathcal L:\left[t_0,t_1\right]\times E^2\to\R de classe C. Ses trois variables étant notées t,x,\dot x (ce qui risque de prê

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https://fr.wikipedia.org/wiki/%C3%89quation_d'Euler-Lagrange

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Blow-up, partial regularity and turbulence in incompressible fluid dynamics

Silja Noëmi Aline Haffter

Weak solutions arise naturally in the study of the Navier-Stokes and Euler equations both from an abstract regularity/blow-up perspective and from physical theories of turbulence. This thesis studies the structure and size of singular set of such weak solutions to equations of incompressible fluid dynamics from two opposite directions. First, it aims to single-out new mechanisms which allow to break the typically supercritical scaling of the equations and, in this way, prevent the formation of singularities either globally or locally in spacetime. Second, in the absence of such mechanisms, we seek to quantify how singular (in terms of dimension of the singular set, for instance) the solutions that we are actually able to construct are. This thesis collects four results pointing in the two directions outlined above which have been obtained in several collaborations during the Ph.D. studies:- a global regularity result for the fractional Navier-Stokes equation slightly blow the critical fractional order,- a global well-posedness result for the defocusing wave equation with slightly supercritical power nonlinearity,- an a.e. smoothness / partial regularity result for the supercritical surface quasigeostrophic (SQG) equation,- an estimate (and a discussion of its sharpness) on the dimension of the singular set of wild Hölder continuous solutions of the incompressible Euler equations.All results presented in the thesis have either been published or are submitted for publication.

EPFL2022

Chargement

Global Fourier spectral methods are excellent tools to solve conserva- tion laws. They enable fast convergence rates and highly accurate solutions. However, being high-order methods, they suffer from the Gibbs phenomenon, which leads to spurious numerical oscillations in the vicinity of discontinuities. This can have a detrimental effect on the solution quality and lead to unphysical results. While local approxima- tion techniques allow for local limiting or reconstruction, there are no such possibilities for global methods. This thesis proposes a neural net- work based method that adds artificial viscosity around discontinuities of the solution to the conservation law. This enables the transforma- tion of discontinuities into steep but continuous jumps. Test cases in one and two spatial dimensions as well as systems of conservation laws (Euler equations) are solved. Furthermore, the method is generalized to other global basis approaches on non-uniform grids and reduced ba- sis methods. The proposed method delivers satisfactory results in all test cases. On the one hand, it is able to detect and handle discontinu- ities. On the other hand, it stays highly accurate for smooth data.

2020

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Non-smooth solutions in incompressible fluid dynamics

Luigi De Rosa

This work is devoted to the study of the main models which describe the motion of incompressible fluids, namely the Navier-Stokes, together with their hypodissipative version, and the Euler equations. We will mainly focus on the analysis of non-smooth weak solutions to those equations. Most of the results have been obtained by using the convex integration techniques introduced by Camillo De Lellis and László Székelyhidi in the context of the Euler equations, which recently led to the proof of the Onsager's conjecture on the anomalous dissipation of the kinetic energy. With various refinements of those iterative schemes we prove ill-posedness of Leray-Hopf weak solutions of the hypodissipative Navier-Stokes equations, sharpness of the kinetic energy regularity for Euler, typicality results in the sense of Baire's category for both Euler and Navier-Stokes, estimate on the dimension of the singular set in time of non-conservative Hölder weak solutions of the Euler equations. Moreover, building on different techniques, we also address some regularizing effects of those equations in various classes of weak solutions with some fractional differentiability in terms of Hölder, Sobolev and Besov regularity. The latter make use of new abstract interpolation results for multilinear operators which we developed for our specific context but which may also have independent interests.

EPFL2021

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