Displaying items by tag: modulus of continuity

Changhui Tan, Liutang Xue

Abstract

We prove global well-posedness of the periodic unidirectional Euler-alignment system in every dimension for supercritical dissipation of order \(0<\alpha<1\) and arbitrary non-vacuum initial data in the Sobolev class of the local theory. The key idea is to propagate simultaneously two moduli of continuity at critical scales for the density \(\rho\) and the potential gradient \(\Gamma=\nabla\Lambda^{-\alpha}u\), both of which have scaling-invariant amplitudes. We also establish exponential alignment of the velocity and exponential convergence of the density to a traveling profile.

This work is supported by NSF grants DMS #2238219

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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