Abstract
The Cucker–Smale flocking model belongs to a wide class of kinetic models that describe a collective motion of interacting particles that exhibit some specific tendency, e.g. to aggregate, flock or disperse. This paper examines the kinetic Cucker–Smale equation with a singular communication weight. Given a compactly supported measure as an initial datum we construct a global in time weak measure-valued solution in the space \({C_{weak}(0,\infty;\mathcal{M})}\). The solution is defined as a mean-field limit of the empirical distributions of particles, the dynamics of which is governed by the Cucker–Smale particle system. The studied communication weight is \({\psi(s)=|s|^{-\alpha}}\) with \({\alpha \in \left(0,\frac 12\right)}\). This range of singularity admits the sticking of characteristics/trajectories. The second result concerns the weak-atomic uniqueness property stating that a weak solution initiated by a finite sum of atoms, i.e. Dirac deltas in the form \({m_i \delta_{x_i} \otimes \delta_{v_i}}\), preserves its atomic structure. Hence these coincide with unique solutions to the system of ODEs associated with the Cucker–Smale particle system.
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Communicated by L. Saint-Raymond
JP was supported by International Ph.D. Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007-2013 funded by the European Regional Development Fund (Ph.D. Programme: Mathematical Methods in Natural Sciences) and partially supported by the Polish NCN Grant PRELUDIUM 2013/09/N/ST1/04113.
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Mucha, P.B., Peszek, J. The Cucker–Smale Equation: Singular Communication Weight, Measure-Valued Solutions and Weak-Atomic Uniqueness. Arch Rational Mech Anal 227, 273–308 (2018). https://doi.org/10.1007/s00205-017-1160-x
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DOI: https://doi.org/10.1007/s00205-017-1160-x