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The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain

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Abstract

We study the global existence and uniqueness of a strong solution to the kinetic thermomechanical Cucker-Smale (for short, TCS) model coupled with Stokes equations in the whole space. The coupled system consists of the kinetic TCS equation for a particle ensemble and the Stokes equations for a fluid via a drag force. In this paper, we present a complete analysis of the existence of global-in-time strong solutions to the coupled model without any smallness restrictions on the initial data.

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References

  1. 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

    Article  ADS  MathSciNet  Google Scholar 

  2. Bae H O, Choi Y P, Ha S Y, Kang M J. Asymptotic flocking dynamics of Cucker-Smale particles immersed in compressible fluids. Discrete Contin Dyn Syst, 2014, 34: 4419–4458

    Article  MathSciNet  Google Scholar 

  3. 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

    Article  ADS  MathSciNet  Google Scholar 

  4. Bae H O, Choi Y P, Ha S Y, Kang M J. Global existence of strong solutions to the Cucker-Smale-Stokes system. J Math Fluid Mech, 2016, 18: 381–396

    Article  MathSciNet  Google Scholar 

  5. 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

    Article  MathSciNet  Google Scholar 

  6. Carrillo J A, Choi Y P, Karper T K. On the analysis of a coupled kinetic-fluid model with local alignment forces. Ann I H Poincare-AN, 2016, 33: 273–307

    Article  ADS  MathSciNet  Google Scholar 

  7. Choi Y P, Kwon B. Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations. Nonlinearity, 2015, 28: 3309–3336

    Article  ADS  MathSciNet  Google Scholar 

  8. Bae H O, Choi Y P, Ha S Y, Kang M J. Global existence of strong solutions to the Cucker-Smale-Stokes system. J Math Fluid Mech, 2016, 18(2): 381–396

    Article  MathSciNet  Google Scholar 

  9. Cucker F, Smale S. Emergent behavior in flocks. IEEE Trans Automat Control, 2007, 52: 852–862

    Article  MathSciNet  Google Scholar 

  10. Choi Y P. Large-time behavior for the Vlasov/compressible Navier-Stokes equations. J Math Phys, 2016, 57: 071501

    Article  ADS  MathSciNet  Google Scholar 

  11. Ha S Y, Ruggeri T. Emergent dynamics of a thermodynamically consistent particle model. Arch Ration Mech Anal, 2017, 223: 1397–1425

    Article  MathSciNet  Google Scholar 

  12. Ha S Y, Kim J, Min C H, et al. A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field. Anal Appl, 2018, 16: 757–805

    Article  MathSciNet  Google Scholar 

  13. Ha S Y, Kim J, Min C H, et al. Uniform stability and mean-field limit of a thermodynamic Cucker-Smale model. Quart Appl Math, 2019, 77: 131–176

    Article  MathSciNet  Google Scholar 

  14. Ha S Y, Kim J, Ruggeri T. Emergent behaviors of thermodynamic Cucker-Smale particles. SIAM J Math Anal, 2018, 50: 3092–3121

    Article  MathSciNet  Google Scholar 

  15. Dong J G, Ha S Y, Kim D, Kim J. Time-delay effect on the flocking in an ensemble of thermomechanical Cucker-Smale particles. J Differential Equations, 2019, 266: 2373–2407

    Article  ADS  MathSciNet  Google Scholar 

  16. Baranger C, Boudin L, Jabin P E, Mancini S. A modeling of biospray for the upper airways. ESAIM Proc, 2005, 14: 41–47

    Article  MathSciNet  Google Scholar 

  17. Sohr H. The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Basel: Springer, 2012

    Google Scholar 

  18. Li F, Mu Y, Wang D. Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Plank equations: global existence near the equilibrium and large time behavior. SIAM J Math Anal, 2017, 49: 984–1026

    Article  MathSciNet  Google Scholar 

  19. Mellet A, Vasseur A. Global weak solutions for a Vlasov-Fokker-Plank/Navier-Stokes system of equations. Math Models Methods Appl Sci, 2007, 17: 1039–1063

    Article  MathSciNet  Google Scholar 

  20. Giga Y, Sohr H. Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J Funct Anal, 1991, 102(1): 72–94

    Article  MathSciNet  Google Scholar 

  21. Hamdache K. Global existence and large time behaviour of solutions for the Vlasov-Stokes equations. Japan J Indust Appl Math, 1998, 15: 51–74

    Article  MathSciNet  Google Scholar 

  22. Kim J, Zou W Y. Solvability and blow-up criterion of the thermomechanical Cucker-Smale-Navier-Stokes. Kinet Relat Models, 2020, 13: 623–651

    Article  MathSciNet  CAS  Google Scholar 

  23. Jin C Y. Global existence of strong solutions to the Kinetic Cucker-Smale model coupled with the stokes equations. J Math Phys, 2021, 62: 021502

    Article  ADS  MathSciNet  Google Scholar 

  24. Mathiaud J. Local smooth solutions of a thin spray model with collisions. Math Models Methods Appl Sci, 2010, 20: 191–221

    Article  MathSciNet  CAS  Google Scholar 

  25. Yu C. Global weak solutions to the incompressible Navier-Stokes-Vlasov equations. J Math Pures Appl, 2013, 100: 275–293

    Article  MathSciNet  Google Scholar 

  26. Vicsek T, Czirók A, Ben-Jacob E, et al. Novel type of phase transition in a system of self-driven particles. Phys Rev Lett, 1995, 75: 1226–1229

    Article  ADS  MathSciNet  CAS  PubMed  Google Scholar 

  27. Vinkovic I, Aguirre C, Simoëns S, Gorokhovski M. Large eddy simulation of droplet dispersion for inhomo-geneous turbulent wall flow. Int J Multiph Flow, 2006, 32: 344–364

    Article  CAS  Google Scholar 

  28. Wang D, Yu C. Global weak solutions to the inhomogeneous Navier-Stokes-Vlasov equations. J Differential Equations, 2014, 259: 3976–4008

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing, 100029, China

    Weiyuan Zou

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  1. Weiyuan Zou
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Correspondence to Weiyuan Zou.

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Conflict of Interest The author declares no conflict of interest.

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This research was supported by the National Natural Science Foundation of China (12001033).

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Zou, W. The global existence of strong solutions to thermomechanical Cucker-Smale-Stokes equations in the whole domain. Acta Math Sci 44, 887–908 (2024). https://doi.org/10.1007/s10473-024-0307-8

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  • DOI: https://doi.org/10.1007/s10473-024-0307-8

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