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

The Cucker-Smale flocking model belongs to a wide class of kinetic modelsthat describe a collective motion of interacting particles that exhibit somespecific tendency e.g. to aggregate, flock or disperse. The paper examines thekinetic Cucker-Smale equation with a singular communication weight. Given acompactly supported measure as an initial datum we construct a global in timeweak measure-valued solution in the space $C_{weak}(0,\infty;\mathcal{M})$. Thesolution is defined as a mean-field limit of the empirical distributions ofparticles, which dynamics is governed by the Cucker-Smale particle system. Thestudied communication weight is $\psi(s)=|s|^{-\alpha}$ with $\alpha \in(0,\frac 12)$. This range of singularity admits sticking ofcharacteristics/trajectories. The second result concerns the weak--atomicuniqueness property stating that a weak solution initiated by a finite sum ofatoms, i.e. Dirac deltas in the form $m_i \delta_{x_i} \otimes \delta_{v_i}$,preserves its atomic structure. Hence they coincide with unique solutions tothe system of ODEs associated with the Cucker-Smale particle system.

The Cucker-Smale equation: singular communication weight, measure-valued solutions and weak-atomic uniqueness