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Global Weak Solutions to a Three-Dimensional Quantum Kinetic-Fluid Model

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Abstract

In this paper, we study a quantum kinetic-fluid model in a three-dimensional torus. This model is a coupling of the Vlasov-Fokker-Planck equation and the compressible quantum Navier-Stokes equations with degenerate viscosity. We establish a global weak solution to this model for arbitrarily large initial data when the pressure takes the form p(ρ) = ργ + pc(ρ), where γ > 1 is the adiabatic coefficient and pc(ρ) satisfies

$${p_c}(\rho ) = \left\{ {\matrix{{ - c{\rho ^{ - 4k}}} \hfill & {{\rm{if}}\,\,\rho \le 1,} \hfill \cr {{\rho ^\gamma }} \hfill & {{\rm{if}}\,\,\,\rho > 1} \hfill \cr } } \right.$$

for k ≥ 4 and some constant c > 0.

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References

  1. 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  MATH  Google Scholar 

  2. Baranger C, Desvillettes L. Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J Hyperbolic Differ Equ, 2006, 3(1): 1–26

    Article  MathSciNet  MATH  Google Scholar 

  3. 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(1): 41–80

    Article  MathSciNet  MATH  Google Scholar 

  4. 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(2/3): 67–74

    Article  MathSciNet  MATH  Google Scholar 

  5. Bresch D, Desjardins B. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J Math Pures Appl, 2006, 86(4): 362–368

    Article  MathSciNet  MATH  Google Scholar 

  6. Bresch D, Desjardins B. On the existence of global weak soutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J Math Pures Appl, 2007, 87(1): 57–90

    Article  MathSciNet  MATH  Google Scholar 

  7. Bresch D, Desjardins B, Lin C-K. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm Partial Differential Equations, 2003, 28(3/4): 843–868

    Article  MathSciNet  MATH  Google Scholar 

  8. Brull S, Méhats F. Derivation of viscous correction terms for the isothermal quantum Euler model. Z Angew Math Mech, 2010, 90(3): 219–230

    Article  MathSciNet  MATH  Google Scholar 

  9. Bürger R, Wendland W L, Concha F. Model equations for gravitational sedimentation-consolidation processes. Z Angew Math Mech, 2000, 80(2): 79–92

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao W, Jiang P. Global bounded weak entropy solutions to the Euler-Vlasov equations in fluid-particle system. SIAM J Math Anal, 2021, 53(4): 3958–3984

    Article  MathSciNet  MATH  Google Scholar 

  11. Carrillo J A, Goudon T. Stability and asymptotic analysis of a fluid-particle interaction model. Comm Partial Differential Equations, 2006, 31(7/9): 1349–1379

    Article  MathSciNet  MATH  Google Scholar 

  12. Chae M, Kang K, Lee J. Global classical solutions for a compressible fluid-particle interaction model. J Hyperbolic Differ Equ, 2013, 10(3): 537–562

    Article  MathSciNet  MATH  Google Scholar 

  13. Choi Y-P, Jung J. Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain. Math Models Methods Appl Sci, 2021, 31(11): 2213–2295

    Article  MathSciNet  MATH  Google Scholar 

  14. Choi Y-P, Jung J. Asymptotic analysis for Vlasov-Fokker-Planck/compressible Navier-Stokes equations with a density-dependent viscosity. AIMS Ser Appl Math, 2020, 10: 145–163

    MathSciNet  MATH  Google Scholar 

  15. Duan R, Liu S. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinet Relat Models, 2013, 6(4): 687–700

    Article  MathSciNet  MATH  Google Scholar 

  16. Gisclon M, Lacroix-Violet I. About the barotropic compressible quantum Navier-Stokes equations. Nonlinear Anal, 2015, 128: 106–121

    Article  MathSciNet  MATH  Google Scholar 

  17. Jüngel A. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J Math Anal, 2010, 42(3): 1025–1045

    Article  MathSciNet  MATH  Google Scholar 

  18. Jüngel A. Effective velocity in compressible Navier-Stokes equations with third-order derivatives. Nonlinear Anal, 2011, 74(8): 2813–2818

    Article  MathSciNet  MATH  Google Scholar 

  19. Karper T K, Mellet A, Trivisa K. Existence of weak solutions to kinetic flocking models. SIAM J Math Anal, 2013, 45(1): 215–243

    Article  MathSciNet  MATH  Google Scholar 

  20. Lacroix-Violet I, Vasseur A. Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit. J Math Pures Appl, 2018, 114(9): 191–210

    Article  MathSciNet  MATH  Google Scholar 

  21. Li F, Li Y. Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain. Commun Pure Appl Anal, 2021, 20(10): 3583–3604

    Article  MathSciNet  MATH  Google Scholar 

  22. Li F, Li Y, Sun B. Global weak solutions and asymptotic analysis for a kinetic-fluid model with a heterogeneous friction force. preprint

  23. Li F, Mu Y, Wang D. Strong solutions to the compressible Navier-Stokes-Vlasov-Fokker-Planck equations: global existence near the equilibrium and large time behavior. SIAM J Math Anal, 2017, 49(2): 984–1026

    Article  MathSciNet  MATH  Google Scholar 

  24. Li H-L, Shou L-Y. Global well-posedness of one-dimensional compressible Navier-Stokes-Vlasov system. J Differential Equations, 2021, 280: 841–890

    Article  MathSciNet  MATH  Google Scholar 

  25. Li H-L, Shou L-Y. Global weak solutions for compressible Navier-Stokes-Vlasov-Fokker-Planck system. Commun Math Res, 2023, 39(1): 136–172

    Article  MathSciNet  MATH  Google Scholar 

  26. Li Y. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system with nonhomogeneous boundary data. Z Angew Math Phys, 2021, 72 (2): Art. 51

  27. Li Y, Sun B. Global weak solutions to a quantum kinetic-fluid model with large initial data. Nonlinear Analysis: Real World Applications, 2023, 71: Art 103822

  28. Mellet A, Vasseur A. Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math Models Methods Appl Sci, 2007, 17(7): 1039–1063

    Article  MathSciNet  MATH  Google Scholar 

  29. Mellet A, Vasseur A. Asymptotic analysis for a Vlasov-Fokker-Planck/compressible Navier-Stokes system of equations. Comm Math Phys, 2008, 281(3): 573–596

    Article  MathSciNet  MATH  Google Scholar 

  30. Mucha P-B, Pokorný M, Zatorska E. Chemically reacting mixtures of degenerated parabolic setting. J Math Phys, 2013, 54(7): 311–341

    Article  MathSciNet  MATH  Google Scholar 

  31. Oron A, Davis S-H, Bankoff S-G. Long-scale evolution of thin liquid films. Rev Mod Phys, 1997, 69: 931–980

    Article  Google Scholar 

  32. Sartory W K. Three-component analysis of blood sedimentation by the method of characteristics. Math Biosci, 1977, 33(1/2): 145–165

    Article  MATH  Google Scholar 

  33. Simon J. Compact sets in the space Lp(0, T; B). Ann Math Pure Appl, 1986, 146: 65–96

    Article  Google Scholar 

  34. Spannenberg A, Galvin K P. Continuous differential sedimentation of a binary suspension. Chem Engrg Aust, 1996, 21: 7–11

    Google Scholar 

  35. Tang T, Niu C. Global existence of weak solutions to the quantum Navier-Stokes equations (in Chinese). Acta Mathematica Scientia, 2022, 42A(2): 387–400

    MATH  Google Scholar 

  36. Vasseur A, Yu C. Global weak solutions to the compressible quantum Navier-Stokes equations with damping. SIAM J Math Anal, 2016, 48(2): 1489–1511

    Article  MathSciNet  MATH  Google Scholar 

  37. Williams F A. Spray combustion and atomization. Physics of Fluids, 1958, 1(6): 541–555

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Fucai Li & Yue Li

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Baoyan Sun

Authors
  1. Fucai Li
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  2. Yue Li
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Correspondence to Yue Li.

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Conflict of Interest

The authors declare no conflict of interest.

F. Li and Y. Li’s research were supported by the NSFC (12071212). And F. Li’s research was also supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. B. Sun’s research was supported by NSFC (12171415) and the Scientific Research Foundation of Yantai University (2219008).

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Li, F., Li, Y. & Sun, B. Global Weak Solutions to a Three-Dimensional Quantum Kinetic-Fluid Model. Acta Math Sci 43, 2089–2107 (2023). https://doi.org/10.1007/s10473-023-0510-z

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  • DOI: https://doi.org/10.1007/s10473-023-0510-z

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