Generalized compressible flows and solutions of the H(div) geodesic problem
| Autor: | Thomas Gallouët, Andrea Natale, François-Xavier Vialard |
|---|---|
| Přispěvatelé: | Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire d'Informatique Gaspard-Monge (LIGM), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, ANR-16-CE40-0014,MAGA,Monge-Ampère et Géométrie Algorithmique(2016), European Project: 609102,EC:FP7:PEOPLE,FP7-PEOPLE-2013-COFUND,PRESTIGE(2014), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL) |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Geodesic Space (mathematics) 01 natural sciences symbols.namesake Mathematics - Analysis of PDEs Mathematics (miscellaneous) FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] Mathematics - Numerical Analysis 0101 mathematics Mathematics Probability measure Mechanical Engineering 010102 general mathematics Numerical Analysis (math.NA) 010101 applied mathematics Cone (topology) Euler's formula symbols Diffeomorphism Relaxation (approximation) Solving the geodesic equations Analysis [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Analysis of PDEs (math.AP) |
| Zdroj: | Archive for Rational Mechanics and Analysis Archive for Rational Mechanics and Analysis, Springer Verlag, In press, ⟨10.1007/s00205-019-01453-x⟩ Archive for Rational Mechanics and Analysis, In press, ⟨10.1007/s00205-019-01453-x⟩ |
| ISSN: | 0003-9527 1432-0673 |
| Popis: | We study the geodesic problem on the group of diffeomorphism of a domain $$M\subset {\mathbb {R}}^d$$, equipped with the $$H(\mathrm {div})$$ metric. The geodesic equations coincide with the Camassa–Holm equation when $$d=1$$, and represent one of its possible multi-dimensional generalizations when $$d>1$$. We propose a relaxation a la Brenier of this problem, in which solutions are represented as probability measures on the space of continuous paths on the cone over M. We use this relaxation to prove that smooth $$H(\mathrm {div})$$ geodesics are globally length minimizing for short times. We also prove that there exists a unique pressure field associated to solutions of our relaxation. Finally, we propose a numerical scheme to construct generalized solutions on the cone and present some numerical results illustrating the relation between the generalized Camassa–Holm and incompressible Euler solutions. |
| Databáze: | OpenAIRE |
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