68 research outputs found
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
Local well-posedness of the generalized Cucker-Smale model
In this paper, we study the local well-posedness of two types of generalized
Cucker-Smale (in short C-S) flocking models. We consider two different
communication weights, singular and regular ones, with nonlinear coupling
velocities for . For the singular
communication weight, we choose with and in dimension . For the regular case, we
select belonging to (L_{loc}^\infty \cap
\mbox{Lip}_{loc})(\mathbb{R}^d) and . We also
remark the various dynamics of C-S particle system for these communication
weights when
On the Lagrangian Trajectories for the One-Dimensional Euler Alignment Model without Vacuum Velocity
A well-known result of Carrillo, Choi, Tadmor, and Tan states that the 1D
Euler Alignment model with smooth interaction kernels possesses a 'critical
threshold' criterion for the global existence or finite-time blowup of
solutions, depending on the global nonnegativity (or lack thereof) of the
quantity . In this note, we rewrite the 1D
Euler Alignment model as a first-order system for the particle trajectories in
terms of a certain primitive of ; using the resulting structure,
we give a complete characterization of global-in-time existence versus
finite-time blowup of regular solutions that does not require a velocity to be
defined in the vacuum. We also prove certain upper and lower bounds on the
separation of particle trajectories, valid for smooth and weakly singular
kernels, and we use them to weaken the hypotheses of Tan sufficient for the
global-in-time existence of a solution in the weakly singular case, when the
order of the singularity lies in the range .Comment: 12 pages, 0 figures. Rewritten manuscript corrects a faulty
conclusion of the previous versio
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