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 note we continue our study of unidirectional solutions tohydrodynamic Euler alignment systems with strongly singular communicationkernels $\phi(x):=|x|^{-(n+\alpha)}$ for $\alpha\in(0,2)$. Here, we considerthe critical case $\alpha=1$ and establish a couple of global existence resultsof smooth solutions, together with a full description of their long timedynamics. The first one is obtained via Schauder-type estimates under a nullinitial entropy condition and the other is a small data result. In fact, usingDuhamel's approach we get that any solution is almost Lipschitz-continuous inspace. We extend the notion of weak solution for $\alpha\in[1,2)$ and prove theexistence of global Leray-Hopf solutions. Furthermore, we give an anisotropicOnsager-type criteria for the validity of the natural energy law for weaksolutions of the system. Finally, we provide a series of quantitative estimatesthat show how far the density of the limiting flock is from a uniformdistribution depending solely on the size of the initial entropy.

Global existence and limiting behavior of unidirectional flocks for the fractional Euler Alignment system