Displaying items by tag: shock formation

Thomas Hamori and Changhui Tan

Nonlinear Analysis: Real World Applications, Volume 73, 103899, (2023).

Abstract

We study a class of traffic flow models with nonlocal look-ahead interactions. The global regularity of solutions depend on the initial data. We obtain sharp critical threshold conditions that distinguish the initial data into a trichotomy: subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams.

	doi:10.1016/j.nonrwa.2023.103899
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264
	This work is supported by a UofSC VPR ASPIRE I grant

Manas Bhatnagar, Hailiang Liu and Changhui Tan

Journal of Differential Equations, Volume 375, pp. 82-119 (2023)

Abstract

This paper is concerned with the global wellposedness of the Euler-Poisson-alignment (EPA) system. This system arises from collective dynamics, and features two types of nonlocal interactions: the repulsive electric force and the alignment force. It is known that the repulsive electric force generates oscillatory solutions, which is difficult to be controlled by the nonlocal alignment force using conventional comparison principles. We construct invariant regions such that the solution trajectories cannot exit, and therefore obtain global wellposedness for subcritical initial data that lie in the invariant regions. Supercritical regions of initial data are also derived which leads to finite-time singularity formations. To handle the oscillation and the nonlocality, we introduce a new way to construct invariant regions piece by piece in the phase plane of a reformulation of the EPA system. Our result is extended to the case when the alignment force is weakly singular. The singularity leads to the loss of a priori bounds crucial in our analysis. With the help of improved estimates on the nonlocal quantities, we design non-trivial invariant regions that guarantee global wellposedness of the EPA system with weakly singular alignment interactions.

	doi:10.1016/j.jde.2023.07.049
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	This work is supported by NSF grants DMS #1853001, DMS #2108264 and DMS 2238219