Displaying items by tag: global regularity

Alexander Kiselev, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 50, No 6, pp. 6208–6229 (2018).

Abstract

The Euler–Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attrac- tion/repulsion as well as alignment forces. We consider one-dimensional EPA system with a class of singular alignment terms as well as natural attraction/repulsion terms. The singularity of the alignment kernel produces an interesting effect regularizing the solutions of the equation and leading to global regularity for wide range of initial data. This was recently observed in [Do et al., Arch. Ration. Mech. Anal., 228 (2018), pp. 1–37]. Our goal in this paper is to generalize the result and to incorporate the attractive/repulsive potential. We prove that global regularity persists for these more general models.

	doi:10.1137/17M1141515
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	This work is supported by NSF grant DMS #1853001

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.

	doi:10.1007/s00205-017-1184-2
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