An inverse problem for multi-dimensional piston models with large velocity variations

Hu D, Li Q, Zhang Y (2026)

Publication Type: Journal article

Publication year: 2026

Journal

Mathematische Annalen Springer Verlag (Germany)

Book Volume: 395

Article Number: 85

Journal Issue: 4

DOI: 10.1007/s00208-026-03468-8

Abstract

In this paper, we investigate the inverse problem of reconstructing the piston motion and the associated flow field from a prescribed leading shock trajectory and given quiescent constant-density initial data for a spherical piston model in the low-initial-density regime. For a uniformly expanding leading shock, we find that the flow immediately behind the shock exhibits vanishing density and convergence of the velocity to the shock speed as the initial density tends to zero, and we further provide the asymptotic behavior of the entire self-similar flow field through a monotonicity analysis. Beyond the uniformly expanding case, we demonstrate that as long as the leading shock expands with a uniformly positive and bounded speed, has an initial constant-speed phase, and its subsequent acceleration is controlled by a positive power of the initial density in a time-weighted sense, the inverse problem admits a global-in-time piecewise smooth solution for sufficiently small initial density. The analysis relies on the method of characteristics and on quantitative characterizations of the separations between the flow velocity, the shock speed, and the characteristic speeds of the system.

Authors with CRIS profile

Qianfeng Li Lehrstuhl für Analysis

Involved external institutions

East China University of Science and Technology (ECUST) CN China (CN) Fudan University / 复旦大学 CN China (CN)

How to cite

APA:

Hu, D., Li, Q., & Zhang, Y. (2026). An inverse problem for multi-dimensional piston models with large velocity variations. Mathematische Annalen, 395(4). https://doi.org/10.1007/s00208-026-03468-8

MLA:

Hu, Dian, Qianfeng Li, and Yongqian Zhang. "An inverse problem for multi-dimensional piston models with large velocity variations." Mathematische Annalen 395.4 (2026).

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