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/

Zendy Plus

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

Zendy Tools

Zendy Open

We develop a version of Haar and Holmgren methods which applies todiscontinuous solutions of nonlinear hyperbolic systems and allows us tocontrol the L1 distance between two entropy solutions. The main difficulty isto cope with linear hyperbolic systems with discontinuous coefficients. Ourmain observation is that, while entropy solutions contain compressive shocksonly, the averaged matrix associated with two such solutions has compressive orundercompressive shocks, but no rarefaction-shocks -- which are recognized as asource for non-uniqueness and instability. Our Haar-Holmgren-type method restson the geometry associated with the averaged matrix and takes into accountadjoint problems and wave cancellations along generalized characteristics. Itextends the method proposed earlier by LeFloch et al. for genuinely nonlinearsystems. In the present paper, we cover solutions with small total variationand a class of systems with general flux that need not be genuinely nonlinearand includes for instance fluid dynamics equations. We prove that solutionsgenerated by Glimm or front tracking schemes depend continuously in the L1 normupon their initial data, by exhibiting an L1 functional controling the distancebetween two solutions.

Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems