Global Regularity for the Fractional Euler Alignment System

Abstract

© 2017 Springer-Verlag GmbH Germany We study a pressureless Euler system with a non-linear density-dependent alignment term, originating in the Cucker–Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density dependence is natural: the alignment rate increases in the areas of high density due to species discomfort. The diffusive term has the order of a fractional Laplacian (Formula presented.). The corresponding Burgers equation with a linear dissipation of this type develops shocks in a finite time. We show that the alignment nonlinearity enhances the dissipation, and the solutions are globally regular for all (Formula presented.). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.