Research
Wednesday, 18 January 2017 23:08

## Global regularity for the fractional Euler alignment system

Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan

Archive for Rational Mechanics and Analysis, Volume 228, No 1, pp. 1-37 (2018).

Abstract

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 $$(-\partial_{xx})^{\alpha/2}, \alpha\in(0,1)$$. 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 $$\alpha\in(0,1)$$. To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.