English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

It was recently proven by De Lellis, Kappeler, and Topalov that the periodicCauchy problem for the Camassa-Holm equations is locally well-posed in thespace Lip (T) endowed with the topology of H^1 (T). We prove here that theLagrangian flow of these solutions are analytic with respect to time and smoothwith respect to the initial data. These results can be adapted to thehigher-order Camassa-Holm equations describing the exponential curves of themanifold of orientation preserving diffeomorphisms of T using the Riemannianstructure induced by the Sobolev inner product H^l (T), for l in N, l > 1 (theclassical Camassa-Holm equation corresponds to the case l=1): the periodicCauchy problem is locally well-posed in the space W^{2l-1,infty} (T) endowedwith the topology of H^{2l-1} (T) and the Lagrangian flows of these solutionsare analytic with respect to time with values in W^{2l-1,infty} (T) and smoothwith respect to the initial data. These results extend some earlier resultswhich dealt with more regular solutions. In particular our results cover thecase of peakons, up to the first collision.

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations