# American Institute of Mathematical Sciences

March  2020, 19(3): 1233-1256. doi: 10.3934/cpaa.2020057

## Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime

 1 Department of Mathematics and Research Institute of Natural Sciences, Sookmyung Women's University, Seoul 04310, Republic of Korea 2 Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Republic of Korea 3 Institute of New Media and Communications, Seoul National University, Seoul 08826, Republic of Korea 4 Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author

Received  October 2018 Revised  August 2019 Published  November 2019

We present a hydrodynamic limit from the kinetic thermomechanical Cucker-Smale (TCS) model to the hydrodynamic Cucker-Smale (CS) model in a strong local alignment regime. For this, we first provide a global existence of weak solution, and flocking dynamics for classical solution to the kinetic TCS model with local alignment force. Then we consider one-parameter family of well-prepared initial data to the kinetic TCS model in which the temperature tends to common constant value determined by initial datum, as singular parameter $\varepsilon$ tends to zero. In a strong local alignment regime, the limit model is the hydrodynamic CS model in [8]. To verify this hydrodynamic limit rigorously, we adopt the technique introduced in [5] which combines the relative entropy method together with the 2-Wasserstein metric.

Citation: Moon-Jin Kang, Seung-Yeal Ha, Jeongho Kim, Woojoo Shim. Hydrodynamic limit of the kinetic thermomechanical Cucker-Smale model in a strong local alignment regime. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1233-1256. doi: 10.3934/cpaa.2020057
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