Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model

Ha, Seung-Yeal; Kim, Jeongho; Min, Chan Ho; Ruggeri, Tommaso; Zhang, Xiongtao

doi:10.1090/qam/1517

Authors: Seung-Yeal Ha, Jeongho Kim, Chan Ho Min, Tommaso Ruggeri and Xiongtao Zhang
Journal: Quart. Appl. Math. 77 (2019), 131-176
MSC (2010): Primary 70F99, 92B25
DOI: https://doi.org/10.1090/qam/1517
Published electronically: September 5, 2018
MathSciNet review: 3897922
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Abstract: We present a uniform-in-time stability and uniform mean-field limit of a thermodynamic Cucker-Smale model with small diffusion velocityf (for short, the SDV-TCS model). The original Cucker-Smale model deals with flocking dynamics of mechanical particles, in which the position and momentum are only macroscopic observables. Thus, the original Cucker-Smale model cannot describe some thermodynamic phenomena resulting from the temperature variations among particles and internal variables not taken into account. In [SIAM J. Math. Anal. 50 (2018), pp. 3092–3121] and [Arch. Rational. Mech. Anal. 223 (2017), pp. 1397–1425], a new thermodynamically consistent particle model was proposed from the system of gas mixtures in a rational way. In this paper, we discuss two issues for the SDV-TCS model. First we present a uniform stability of the SDV-TCS model with respect to initial data in the sense that the distance between two solutions is uniformly bounded by that of initial data in a mixed Lebesgue norm. Second, we derive a uniform mean-field limit from the SDV-TCS model to the Vlasov-type kinetic equation for some class of initial data whose empirical measure approximation guarantees exponential flocking in the SDV-TCS model.

References

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