GLOBAL WEAK SOLUTION TO COMPRESSIBLE NAVIER-STIKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE

Fengxia Liu; Boling Guo

doi:10.11948/20200487

2022 Volume 12 Issue 1

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Fengxia Liu, Boling Guo. GLOBAL WEAK SOLUTION TO COMPRESSIBLE NAVIER-STIKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 31-49. doi: 10.11948/20200487

Citation:

Fengxia Liu, Boling Guo. GLOBAL WEAK SOLUTION TO COMPRESSIBLE NAVIER-STIKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 31-49. doi: 10.11948/20200487

GLOBAL WEAK SOLUTION TO COMPRESSIBLE NAVIER-STIKES-LANDAU-LIFSHITZ-MAXWELL EQUATIONS FOR QUANTUM FLUIDS IN DIMENSION THREE

Fengxia Liu^1, ,,
Boling Guo²

1.
Engineering Research Center of Ministry of Education for Financial Computing and Digital Engineering, Renmin University of China, Beijing, 100872, China
2.
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

Corresponding author: Email address: liufengxia91@126.com(F. Liu)

Abstract

Abstract

This paper is concerned with viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell equations in dimension three. We use Faedo-Galerkin method to prove the local existence of weak solution, then combine the a priori estimates to obtain the global existence of solution.
- Viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell system /
- weak solution /
- global solution /
- Faedo-Galerkin method
MSC: 35A01, 35D30, 35Q30

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References

[1]	E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Univer. Press, 2004. Google Scholar
[2]	D. Ferry and J. R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 1993, 48, 7944-7950. doi: 10.1103/PhysRevB.48.7944 CrossRef Google Scholar
[3]	P. Fuchs, A. Juengel and M. V. Renesse, On the lagrangian structure of quantum fluid model, DCDS-Series A, 2017, 34(4), 1375-1396. Google Scholar
[4]	C. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 1994, 54, 409-427. doi: 10.1137/S0036139992240425 CrossRef Google Scholar
[5]	J. Grant, Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations, J. Phys. A, 1973, 6, L151-L153. doi: 10.1088/0305-4470/6/11/001 CrossRef Google Scholar
[6]	M. Gualdani and A. Jungel, Analysis of the viscous quantum hydrodynamic equations for semiconductors, European J. Appl. Math., 2004, 15, 577-595. doi: 10.1017/S0956792504005686 CrossRef Google Scholar
[7]	B. Guo and S. Ding, Landau-Lifshitz-Maxwell Equations, Landau-Lifshitz Equations, 2014. Google Scholar
[8]	B. Guo and F. Liu, Weak and smooth solutions to incompressible Navier-Stokes-Landau-Lifshitz-Maxwell equations, Frontiers of Mathematics in China, 2020, 14(6). Google Scholar
[9]	B. Guo and M. Hong, The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps, Calc. Var., 1993, 1, 311-334. doi: 10.1007/BF01191298 CrossRef Google Scholar
[10]	B. Guo and G. Wang, Global finite weak solution to the viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell model in 2-dimension, Appl. Math., 2016. Google Scholar
[11]	B. Guo and S. Ding, Landau-Lifshitz Equations, Word Science: Singapore, 2008. Google Scholar
[12]	S. Jiang and P. Zhang, Global sphereically symmetric solutions of the compressible isentropic Navier-Stokes equations, Comm. Math. Phys., 2001, 215, 559-581. doi: 10.1007/PL00005543 CrossRef Google Scholar
[13]	A. Jüngel, Global weak solutions to cmpressible Navier-Stokes equations for quantum fluids, SIAM J. Appl. Math., 2010, 42(3), 1025-1045. doi: 10.1137/090776068 CrossRef Google Scholar
[14]	M. Loffredo and L. Morato, On the creation of quantum vortex lines in rotating He Ⅱ, Societ Italiana di Fisica, 1993, 108B, 205-215. Google Scholar
[15]	E. Madelung, Quantentheorie in hydrodynamischer form, Z. Physik, 1927, 40, 322-326. doi: 10.1007/BF01400372 CrossRef Google Scholar
[16]	L. Nirenberg, On elliptic partial differential equations, Anna. della Scuola Normale Superiore di Pisa-Classe di Scienze, 1959, 13(2), 115-162. Google Scholar
[17]	A. Novotny and I. Straškra, Introduction to the mathematical theory of compressible flow, Oxford Univer. Press, 2004. Google Scholar
[18]	R. Wyatt, Quantum Dynamics with Trajectories, Springer, New York, 2005. Google Scholar