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

doi:10.1007/s10473-023-0510-z

Abstract

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

CLC Number:

35Q35

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.

Trendmd

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
[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^*

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Liao Yuankang. Nonlinear Stability of Viscous Shock Waves for One-dimensional Isentropic Compressible Navier-Stokes Equations with Density-Dependent Viscosity [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1149-1169.
[2]	Tong Tang,Cong Niu. Global Existence of Weak Solutions to the Quantum Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2022, 42(2): 387-400.
[3]	Kexin Luo,Shaoyong Lai. Global Weak Solutions to a High-Order Camass-Holm Type Equation [J]. Acta mathematica scientia,Series A, 2022, 42(2): 427-441.
[4]	Daoguo Zhou. Regularity Criteria in Lorentz Spaces for the Three Dimensional Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1396-1404.
[5]	Weilin Zou,Yuanchun Ren,Meipin Xiao. Regularizing Effect of L¹ Interplay Between Coefficients in Nonlinear Degenerate Elliptic Equations [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1405-1414.
[6]	Shangxi Guo. Existence of Global Strong Solutions for Initial Value Problems of One-Dimensional Compressible Navier-Stokes Equations [J]. Acta mathematica scientia,Series A, 2021, 41(3): 642-651.
[7]	Jiao Luo,Qian Qi,Hong Luo. Weak Solutions to Higher-Order Anisotropic Cahn-Hilliard-Navier-Stokes Systems [J]. Acta mathematica scientia,Series A, 2020, 40(6): 1599-1611.
[8]	Kai Li,Han Yang,Fan Wang. Study on Weak Solution and Strong Solution of Incompressible MHD Equations with Damping in Three-Dimensional Systems [J]. Acta mathematica scientia,Series A, 2019, 39(3): 518-528.
[9]	Zhang Shengui. Infinitely Many Solutions for a Bi-Nonlocal Problem Involving p(x)-Laplacian-Like Operator [J]. Acta mathematica scientia,Series A, 2018, 38(3): 514-526.
[10]	Xing Chao, Ren Mengzhang, Luo Hong. The Existence of Global Weak Solutions to Thermohaline Circulation Equations [J]. Acta mathematica scientia,Series A, 2018, 38(2): 284-290.
[11]	Du Guangwei, Niu Pengcheng. Higher Integrability for Very Weak Solutions of Obstacle Problems to Nonlinear Subelliptic Equations [J]. Acta mathematica scientia,Series A, 2017, 37(1): 122-145.
[12]	YE Yu-Lin, DOU Chang-Sheng, JIU Quan-Sen. LOCAL WELL-POSEDNESS TO THE CAUCHY PROBLEM OF THE 3-D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY [J]. Acta mathematica scientia,Series A, 2014, 34(3): 851-871.
[13]	LIU Ying, LI Jia. L² Decay for Weak Solutions of the MHD Equations in Half Space [J]. Acta mathematica scientia,Series A, 2010, 30(4): 1166-1175.
[14]	ZHENG Shen-Zhou, WANG Xi-Fen. Regularity of Very Weak Solutions for a Class of Quasilinear Subelliptic Equations [J]. Acta mathematica scientia,Series A, 2010, 30(2): 432-439.
[15]	Ran Qikang. Existence of Weak Solutions to a Class of Elliptic Stochastic Partial Differential Equations [J]. Acta mathematica scientia,Series A, 2008, 28(2): 320-328.