Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8256193 | Physica D: Nonlinear Phenomena | 2018 | 7 Pages |
Abstract
We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel Ï(x)=|x|â(1+α). Following our works Shvydkoy and Tadmor (2017) [1], [2] which focused on the range 1â¤Î±<2, and Do et al. (2017) which covered the range 0<α<1, in this paper we revisit the latter case and give a short(-er) proof of global in time existence of smooth solutions, together with a full description of their long time dynamics. Specifically, we prove that starting from any initial condition in (Ï0,u0)âH2+αÃH3, the solution approaches exponentially fast to a flocking state solution consisting of a wave ÏÌ=Ïâ(xâtuÌ) traveling with a constant velocity determined by the conserved average velocity uÌ. The convergence is accompanied by exponential decay of all higher order derivatives of u.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Roman Shvydkoy, Eitan Tadmor,