OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS

doi:10.1007/s10473-025-0221-8

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 659-683.doi: 10.1007/s10473-025-0221-8

OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS

Houzhi Tang¹, Shuxing Zhang², Weiyuan Zou^3,*

1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China;
2. School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China;
3. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

收稿日期:2023-11-02 修回日期:2024-05-27 出版日期:2025-03-25 发布日期:2025-05-08

OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS

Houzhi Tang¹, Shuxing Zhang², Weiyuan Zou^3,*

1. School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China;
2. School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China;
3. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China

Received:2023-11-02 Revised:2024-05-27 Online:2025-03-25 Published:2025-05-08
Contact: *Weiyuan Zou, E-mail: zwy@amss.ac.cn
About author:Houzhi Tang, E-mail: houzhitang@ahnu.edu.cn;Shuxing Zhang, E-mail: sxzhang@amss.ac.cn
Supported by:
The first author's work was supported by the Anhui Provincial Natural Science Foundation (2408085QA031); the third author's work was supported by the National Natural Science Foundation of China (12001033).

摘要/Abstract

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 $\mathbb{R}^3$ . 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.

关键词: Cucker-Smale model, incompressible Navier-Stokes equations, weak solution, large-time behavior, optimal decay rates

Abstract: 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 $\mathbb{R}^3$ . 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.

Key words: Cucker-Smale model, incompressible Navier-Stokes equations, weak solution, large-time behavior, optimal decay rates

中图分类号:

35B40

Houzhi Tang, Shuxing Zhang, Weiyuan Zou. OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS[J]. 数学物理学报(英文版), 2025, 45(2): 659-683.

Houzhi Tang, Shuxing Zhang, Weiyuan Zou. OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS[J]. Acta mathematica scientia,Series B, 2025, 45(2): 659-683.

参考文献

[1] Boudin L, Desvillettes L, Grandmont C, Moussa A. Global existence of solution for the coupled Vlasov and Navier-Stokes equations. Differential Integral Equations, 2009, 22: 1247-1271
[2] Bae H O, Choi Y P, Ha S Y, Kang M J. Time-asymptotic interaction of flocking particles and incompressible viscous fluid. Nonlinearity, 2012, 25: 1155-1177
[3] Bae H O, Choi Y P, Ha SY, Kang M J. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete and Continuous Dynamical System, 2014, 34: 4419-4458
[4] Bae H O, Choi Y P, Ha S Y, Kang M J. Global existence of strong solution for the Cucker-Smale-Navier-Stokes system. J Differ Equat, 2014, 257: 2225-2255
[5] Baranger C, Boudin L, Jabin P E, Mancini S. A modeling of biospray for the upper airways. ESAIM Proc, 2005, 14: 41-47
[6] Berres S, Bürger R, Karlsen K H, Tory E M. Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J Appl Math, 2003, 64 41-80
[7] Berres S, Bürger R, Tory E M. Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixtures. Comput Vis Sci, 2004, 6: 67-74
[8] Bürger R, Wendland W L, Concha F. Model equations for gravitational sedimentation-consolidation processes. Z Angew Math Mech, 2000, 80: 79-92
[9] Boudin L, Desvillettes L, Grandmont C, Moussa A. Global existence of solutions for the coupled Vlasov and Navier-Stokes equations . Diff Integral Eqns, 2009, 22(11/12): 1247-1271
[10] Choi Y P, Lee J. Global existence of weak and strong solutions to Cucker-Smale-Navier-Stokes equations in

$\mathbb{R}^2$ . Nonlinear Analysis: Real World Applications, 2016, 27: 158-182
[11] Choi Y P, Ha S Y, Jung J, Kim J. Global dynamics of the thermomechanical Cucker-Smale ensemble immersed in incompressible viscous fluids. Nonlinearity, 2019, 32: 1597-1640
[12] Choi Y P, Kwon B. Global well-posedness and large-time behavior for the inhomogeneous V lasov- N avier- S tokes equations. Nonlinearity, 2015, 28: 3309-3336
[13] Constantin P, Masmoudi N. Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D . Commun Math Phys, 2007, 278(1): 179-191
[14] Cucker F, Smale S. Emergent behavior in flocks. IEEE Trans Automat Control, 2007, 52(5): 852-862
[15] Carrillo J A, Fornasier M, Rosado J, Toscani G. Asymptotic flocking dynamics for the kinetic Cucker-Smale model . SIAM J Math Anal, 2010, 42(1): 218-236
[16] Duan R, Fornasier M, Toscani G. A kinetic flocking model with diffusion. Commun Math Phys, 2010, 300(1): 95-145
[17] Falkovich G, Fouxon A, Stepanov M G. Acceleration of rain initiation by cloud turbulence. Nature, 2002, 219: 151-154
[18] Fujita H, Kato T. On the Navier-Stokes initial value problem. I : Arch Ration Mech Anal,1964, 16(4): 269-315
[19] Ha S Y, Liu J G. A simple proof of the Cucker-Smale flocking dynamics and mean-field limit . Commun Math Sci, 2009, 7(2): 297-325
[20] Ha S Y, Tadmor E. From particle to kinetic and hydrodynamic descriptions of flocking. Kinet Relat Models, 2008, 1(3): 415-435
[21] Han-Kwan D. Large-time behavior of small-data solutions to the Vlasov-Navier-Stokes system on the whole space . Probab Math Phys, 2022, 3: 35-67
[22] Han-Kwan D, Moussa A, Moyano I. Large time behavior of the V lasov- N avier- S tokes system on the torus . Arch Ration Mech Anal,2020, 3: 1273-1323
[23] Hamdache K. Global existence and large time behavior of solutions for the Vlasov-Stokes equations . Japan J Indust Appl Math, 1998, 15(1): 51-74
[24] Jin C. The local existence and blowup criterion for strong solutions to the kinetic Cucker-Smale model coupled with the compressible Navier-Stokes equations . Nonlinear Anal: RWA,2019, 49: 217-249
[25] Jin C.Global existence and large time behaviors of strong solutions to the Kinetic Cucker-Smale model coupled with the three dimensional incompressible Navier-Stokes equations . arXiv:2112.12900
[26] Kim J, Zou W Y. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes. Kinet Relat Models, 2020, 13: 623-651
[27] Lin F, Liu C, Zhang P. On a micro-macro model for polymeric fluids near equilibrium. Commun Pure Appl Math, 2007, 60(6): 838-866
[28] Lin F, Zhang P, Zhang Z. On the global existence of the smooth solution to the 2-D FENE dumbbell model . Commun Math Phys, 2008, 277(2): 531-553
[29] Leray J. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math, 1934, 63: 193-248
[30] Schonbek M.

$L^2$ decay for weak solutions of the Navier-Stokes equations. Arch Rational Mech Anal, 1985, 88: 209-222
[31] Schonbek M. Large time behavior of solutions to the Navier-Stokes equations. Comm Partial Differential Equations, 1986, 11: 733-763
[32] Schonbek M. Lower bounds of rates of decay for solutions to the Navier-Stokesequations. J Amer Math Soc, 1991, 4: 423-449
[33] Williams F A. Combustion Theory . Menlo Park: Benjamin Cummings, 1985
[34] Zou W. The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain. Acta Math Sci, 2024, 44B(3): 887-908

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OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS

OPTIMAL DECAY RATES FOR THE WEAK SOLUTIONS OF THE FLOCKING PARTICLES COUPLED WITH INCOMPRESSIBLE VISCOUS FLUID MODELS

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