Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication

Young-Pil Choi; Seung-Yeal Ha; Jeongho Kim; Young-Pil Choi; Seung-Yeal Ha; Jeongho Kim

doi:10.3934/nhm.2018017

Networks and Heterogeneous Media

2018, Volume 13, Issue 3: 379-407. doi: 10.3934/nhm.2018017

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Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication

1.
Department of Mathematics and Institute of Applied Mathematics, Inha University, Incheon 22212, Korea
2.
Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea
3.
Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea
4.
Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

Received: 01 September 2017 Revised: 01 October 2017
70F99, 92B25

We study dynamical behaviors of the ensemble of thermomechanical Cucker-Smale (in short TCS) particles with singular power-law communication weights in velocity and temperatures. For the particle TCS model, we present several sufficient frameworks for the global regularity of solution and a finite-time breakdown depending on the blow-up exponents in the power-law communication weights at the origin where the relative spatial distances become zero. More precisely, when the blow-up exponent in velocity communication weight is greater than unity and the blow-up exponent in temperature communication weights is more than twice of blow-up exponent in velocity communication, we show that there will be no finite time collision between particles, unless there are collisions initially. In contrast, when the blow-up exponent of velocity communication weight is smaller than unity, we show that there can be a collision in finite time. For the kinetic TCS equation, we present a local-in-time existence of a unique weak solution using the suitable regularization and compactness arguments.
- Collision,
- flocking,
- the thermomechanical Cucker-Smale model,
- flocking model,
- thermodynamics
Citation: Young-Pil Choi, Seung-Yeal Ha, Jeongho Kim. Propagation of regularity and finite-time collisions for the thermomechanical Cucker-Smale model with a singular communication[J]. Networks and Heterogeneous Media, 2018, 13(3): 379-407. doi: 10.3934/nhm.2018017

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Abstract

We study dynamical behaviors of the ensemble of thermomechanical Cucker-Smale (in short TCS) particles with singular power-law communication weights in velocity and temperatures. For the particle TCS model, we present several sufficient frameworks for the global regularity of solution and a finite-time breakdown depending on the blow-up exponents in the power-law communication weights at the origin where the relative spatial distances become zero. More precisely, when the blow-up exponent in velocity communication weight is greater than unity and the blow-up exponent in temperature communication weights is more than twice of blow-up exponent in velocity communication, we show that there will be no finite time collision between particles, unless there are collisions initially. In contrast, when the blow-up exponent of velocity communication weight is smaller than unity, we show that there can be a collision in finite time. For the kinetic TCS equation, we present a local-in-time existence of a unique weak solution using the suitable regularization and compactness arguments.

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