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Global Regularity for the Fractional Euler Alignment System

  • Tam Do
  • Alexander Kiselev
  • Lenya Ryzhik
  • Changhui Tan
Article

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.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.Department of MathematicsStanford UniversityStanfordUSA
  4. 4.Department of MathematicsRice UniversityHoustonUSA

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