Blowup dynamics for the Euler-Smoluchowski-Poisson equation: Quantized collapse and subvortex interactions

Elio Espejo; Takashi Suzuki; Elio Espejo; Takashi Suzuki

doi:10.3934/math.2026209

AIMS Mathematics

2026, Volume 12, Issue 2: 5120-5151. doi: 10.3934/math.2026209

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Blowup dynamics for the Euler-Smoluchowski-Poisson equation: Quantized collapse and subvortex interactions

Elio Espejo ^{1
,
,},
Takashi Suzuki ²

1.
School of Mathematical Science, The University of Nottingham, Ningbo China
2.
Center for Mathematical Modeling and Data Science, University of Osaka, Japan

Received: 22 November 2025 Revised: 30 January 2026 Accepted: 11 February 2026 Published: 27 February 2026
MSC : 35Q82, 82B21

We investigate the relaxation dynamics of point vortices described by the Euler-Smoluchowski-Poisson equation. Our study reveals a quantization mechanism for collapse masses and demonstrates that the Hamiltonian of the point vortex system governs the dynamics of subcollapses, both in finite-time and infinite-time blowup scenarios. By analyzing the interplay between the Euler term and the Smoluchowski-Poisson structure, we establish that the quantized blowup mechanism and recursive hierarchy observed in the absence of the Euler term persist when it is introduced. Furthermore, we clarify the role of the Euler term in shaping the microscopic dynamics of subcollapses during blowup. Our results extend the understanding of blowup phenomena in this system and highlight the robustness of its underlying geometric and analytical structures.
- Euler-Smoluchowski-Poisson equation,
- blowup of the solution,
- system of point vortices,
- Onsager's theory,
- recursive hierarchy
Citation: Elio Espejo, Takashi Suzuki. Blowup dynamics for the Euler-Smoluchowski-Poisson equation: Quantized collapse and subvortex interactions[J]. AIMS Mathematics, 2026, 12(2): 5120-5151. doi: 10.3934/math.2026209

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Abstract

We investigate the relaxation dynamics of point vortices described by the Euler-Smoluchowski-Poisson equation. Our study reveals a quantization mechanism for collapse masses and demonstrates that the Hamiltonian of the point vortex system governs the dynamics of subcollapses, both in finite-time and infinite-time blowup scenarios. By analyzing the interplay between the Euler term and the Smoluchowski-Poisson structure, we establish that the quantized blowup mechanism and recursive hierarchy observed in the absence of the Euler term persist when it is introduced. Furthermore, we clarify the role of the Euler term in shaping the microscopic dynamics of subcollapses during blowup. Our results extend the understanding of blowup phenomena in this system and highlight the robustness of its underlying geometric and analytical structures.

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