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

In this paper, we study the regularity properties of bounded entropysolutions to the isentropic Euler equations with $\gamma = 3$. First, we use ablow-up technique to obtain a new trace theorem for all such solutions. Second,we use a modified De Giorgi type iteration on the kinetic formulation to show anew partial regularity result on the Riemann invariants. We are able toconclude that in fact for any bounded entropy solution $u$, the density $\rho$is almost everywhere upper semicontinuous away from vacuum. To our knowledge,this is the first example of a nonlinear hyperbolic system, which fails to beTemple class, but has the property that generic $L^\infty$ initial data giverise to bounded entropy solutions with a form of near classical regularity.This provides one example that $2\times 2$ hyperbolic systems can possess someof the more striking regularizing effects known to hold generically in thegenuinely nonlinear, multidimensional scalar setting. While we are not able touse our regularity results to show unconditional uniqueness, the resultssubstantially lower the likelihood that current methods of convex integrationcan be used in this setting.

Unconditional regularity and trace results for the isentropic Euler equations with $\gamma = 3$