Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0

By Roman Shvydkoy, Eitan Tadmor. Physica A: Statistical Mechanics and its Applications, 2017

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Abstract We continue our study of hydrodynamic models of self-organized evolution of agents with singular interaction kernel ϕ ( x ) = | x | − ( 1 + α ) . Following our works Shvydkoy and Tadmor (2017)  [1] , [2] which focused on the range 1 ≤ α 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 ( ρ 0 , u 0 ) ∈ H 2 + α × H 3 , the solution approaches exponentially fast to a flocking state solution consisting of a wave ρ = ρ ∞ ( x − t u ) traveling with a constant velocity determined by the conserved average velocity u . The convergence is accompanied by exponential decay of all higher order derivatives of u .

Subfields: Condensed Matter Physics, Modeling and Simulation, Public Health, Environmental and Occupational Health

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