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<p>This paper studies the global existence and large-time behaviors of weak solutions to the kinetic particle model coupled with the incompressible Navier-Stokes equations in ℝ<sup>3</sup>. First, we obtain the global weak solution using the characteristic and energy methods. Then, under the small assumption of the mass of the particle, we show that the solutions decay at the algebraic time-decay rate. Finally, it is also proved that the above rate is optimal. It should be remarked that if the particle in the coupled system vanishes (i.e. <i>f</i> = 0), our works coincide with the classical results by Schonbek [32] (J Amer Math Soc, 1991, 4: 423–449), which can be regarded as a generalization from a single fluid model to the two-phase fluid one.</p>

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Optimal decay rates for the weak solutions of the flocking particles coupled with incompressible viscous fluid models

  • 期刊论文

  • 发表日期：

    2025年2月6日

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  • Houzhi Tang,
  • Shuxing Zhang,
  • Weiyuan Zou

摘要

This paper studies the global existence and large-time behaviors of weak solutions to the kinetic particle model coupled with the incompressible Navier-Stokes equations in ℝ3. First, we obtain the global weak solution using the characteristic and energy methods. Then, under the small assumption of the mass of the particle, we show that the solutions decay at the algebraic time-decay rate. Finally, it is also proved that the above rate is optimal. It should be remarked that if the particle in the coupled system vanishes (i.e. f = 0), our works coincide with the classical results by Schonbek [32] (J Amer Math Soc, 1991, 4: 423–449), which can be regarded as a generalization from a single fluid model to the two-phase fluid one.