Global Regularity for the Fractional Euler Alignment System

Journal Article (Journal Article)

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 (-∂xx)α/2,α∈(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 α∈ (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.

Full Text

Duke Authors

Cited Authors

  • Do, T; Kiselev, A; Ryzhik, L; Tan, C

Published Date

  • April 1, 2018

Published In

Volume / Issue

  • 228 / 1

Start / End Page

  • 1 - 37

Electronic International Standard Serial Number (EISSN)

  • 1432-0673

International Standard Serial Number (ISSN)

  • 0003-9527

Digital Object Identifier (DOI)

  • 10.1007/s00205-017-1184-2

Citation Source

  • Scopus