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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The paper recalls two of the regularity results for Burgers' equation, anddiscusses what happens in the case of genuinely nonlinear, strictly hyperbolicsystems of conservation laws. The first regularity result which is consideredis Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on thex-derivative of u when u is an entropy solution of the Cauchy problem forBurgers' equation with bounded initial data. Its extensions to the case ofsystems is then mentioned. The second regularity result of debate isSchaeffer's theorem: entropy solutions to Burgers' equation with smooth andgeneric, in a Baire category sense, initial data are piecewise smooth. Thefailure of the same regularity for general genuinely nonlinear systems is nextdescribed. The main focus of this paper is indeed including heuristically anoriginal counterexample where a kind of stability of a shock pattern made byinfinitely many shocks shows up, referring to [Caravenna-Spinolo] for therigorous result.

A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation