Browsing by Author "Tan, C"

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Open Access
Global regularity for 1D Eulerian dynamics with singular interaction forces
(2017-12-18) Kiselev, A; Tan, C
The Euler-Poisson-Alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set agents interacting through mutual attraction/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 the paper by Do, Kiselev, Ryzhik and Tan. 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.
Open Access
Global Regularity for the Fractional Euler Alignment System
(Archive for Rational Mechanics and Analysis, 2017-10-22) Do, T; Kiselev, A; Ryzhik, L; Tan, C
© 2017 Springer-Verlag GmbH Germany 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 (Formula presented.). 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 (Formula presented.). To the best of our knowledge, this is the first example of such regularization due to the non-local nonlinear modulation of dissipation.