GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL*

doi:10.1007/s10473-023-0510-z

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 2089-2107.doi: 10.1007/s10473-023-0510-z

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

Fucai Li¹, Yue Li^1,†, Baoyan Sun²

1. Department of Mathematics, Nanjing University, Nanjing 210093, China;
2. School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

收稿日期:2022-02-14 修回日期:2023-04-30 出版日期:2023-10-26 发布日期:2023-10-25

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

Fucai Li¹, Yue Li^1,†, Baoyan Sun²

1. Department of Mathematics, Nanjing University, Nanjing 210093, China;
2. School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China

Received:2022-02-14 Revised:2023-04-30 Online:2023-10-26 Published:2023-10-25
Contact: †Yue Li, E-mail: liyue2011008@163.com
About author:Fucai Li, E-mail: fli@nju.edu.cn; Baoyan Sun, E-mail: bysun@ytu.edu.cn
Supported by:
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).

摘要/Abstract

摘要： 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(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$.

关键词: kinetic-fluid model, quantum Bohm potential, density-dependent viscosity, weak solutions

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(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$.

Key words: kinetic-fluid model, quantum Bohm potential, density-dependent viscosity, weak solutions

中图分类号:

35Q35

Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*[J]. 数学物理学报, 2023, 43(5): 2089-2107.

Fucai Li, Yue Li, Baoyan Sun. GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*[J]. Acta mathematica scientia,Series A, 2023, 43(5): 2089-2107.

参考文献

[1] Baranger C, Boudin L, Jabin P-E, Mancini S. A modeling of biospray for the upper airways. ESAIM Proc, 2005, 14: 41-47
[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
[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
[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
[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
[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
[7] Bresch D, Desjardins B, Lin C-K. On some compressible fluid models: Korteweg, lubrication, shallow water systems. Comm Partial Differential Equations, 2003, 28(3/4): 843-868
[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
[9] Bürger R, Wendland W L, Concha F. Model equations for gravitational sedimentation-consolidation processes. Z Angew Math Mech, 2000, 80(2): 79-92
[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
[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
[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
[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
[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
[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
[16] Gisclon M, Lacroix-Violet I. About the barotropic compressible quantum Navier-Stokes equations. Nonlinear Anal, 2015, 128: 106-121
[17] Jüngel A. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J Math Anal, 2010, 42(3): 1025-1045
[18] Jüngel A. Effective velocity in compressible Navier-Stokes equations with third-order derivatives. Nonlinear Anal, 2011, 74(8): 2813-2818
[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
[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
[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
[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
[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
[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
[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
[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
[30] Mucha P-B Pokorný M, Zatorska E. Chemically reacting mixtures of degenerated parabolic setting. J Math Phys, 2013, 54(7): 311-341
[31] Oron A, Davis S-H, Bankoff S-G. Long-scale evolution of thin liquid films. Rev Mod Phys, 1997, 69: 931-980
[32] Sartory W K. Three-component analysis of blood sedimentation by the method of characteristics. Math Biosci, 1977, 33(1/2): 145-165
[33] Simon J. Compact sets in the space $L^p(0, T;B)$. Ann Math Pure Appl, 1986, 146: 65-96
[34] Spannenberg A, Galvin K P. Continuous differential sedimentation of a binary suspension. Chem Engrg Aust, 1996, 21: 7-11
[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
[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
[37] Williams F A. Spray combustion and atomization. Physics of Fluids, 1958, 1(6): 541-555

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL^*

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