Quantitative compactness estimates for Hamilton-Jacobi equations

Abstract : We study quantitative compactness estimates in $\mathbf{W}^{1,1}_{loc}$ for the map $S_t$, $t>0$ that associates to every given initial data $u_0\in Lip(\mathbb{R}^N)$ the corresponding solution $S_t u_0$ of a Hamilton-Jacobi equation $u_t+H\big(\nabla_{/!x} u\big)=0\,, \qquad t\geq 0,\quad x\in \mathbb{R}^N,$ with a uniformly convex Hamiltonian $H=H(p)$. We provide upper and lower estimates of order $1/\varepsilon^N$ on the the Kolmogorov $\varepsilon$-entropy in $\mathbf{W}^{1,1}$ of the image through the map $S_t$ of sets of bounded, compactly supported initial data. Estimates of this type are inspired by a question posed by P.D. Lax within the context of conservation laws, andcould provide a measure of the order of "resolution" of a numerical method implemented for this equation.
Keywords :
Type de document :
Pré-publication, Document de travail
31 pages, 1 figure. 2014

Littérature citée [20 références]

https://hal.inria.fr/hal-01098258
Contributeur : Estelle Bouzat <>
Soumis le : mardi 23 décembre 2014 - 14:37:12
Dernière modification le : jeudi 14 juin 2018 - 10:54:02
Document(s) archivé(s) le : mardi 24 mars 2015 - 10:35:30

Fichier

1403.4556v1.pdf
Fichiers produits par l'(les) auteur(s)

Identifiants

• HAL Id : hal-01098258, version 1
• ARXIV : 1403.4556

Citation

Fabio Ancona, Piermarco Cannarsa, Khai T. Nguyen. Quantitative compactness estimates for Hamilton-Jacobi equations. 31 pages, 1 figure. 2014. 〈hal-01098258〉

Métriques

Consultations de la notice

310

Téléchargements de fichiers