Journal
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
Volume 228, Issue 1, Pages 1-37Publisher
SPRINGER
DOI: 10.1007/s00205-017-1184-2
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Funding
- NSF [DMS-1412023, DMS-1311903]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1613603] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1712294] Funding Source: National Science Foundation
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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 . 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 . 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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