November  2019, 24(11): 6141-6166. doi: 10.3934/dcdsb.2019133

Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

2. 

Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing, 100088, China

* Corresponding author: Guangwu Wang

Received  December 2017 Published  November 2019 Early access  July 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China No. 11801107, and the second author is supported by the National Natural Science Foundation of China No. 11731014.

In this paper we investigate the global existence of the weak solutions to the quantum Navier-Stokes-Landau-Lifshitz equations with density dependent viscosity in two dimensional case. We research the model with singular pressure and the dispersive term. The main technique is using the uniform energy estimates and B-D entropy estimates to prove the convergence of the solutions to the approximate system. We also use some convergent theorems in Sobolev space.

Citation: Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, (Second edition), Academic Press, Amsterdam, 2003.

[2]

A. I. Akhiezer, V. G. Yakhtar and S. V. Peletminskii, Spin Waves, North-Holland, 1968.

[3]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. TMA, 18 (1992), 1071-1084.  doi: 10.1016/0362-546X(92)90196-L.

[4]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of quantum Navier-Stokes equations, Arch. Rational Mech. Anal., 225 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[5]

P. Antonelli and S. Spirito, On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations, J. Hyperbolic Differ. Equ., 15 (2018), 133-147, arXiv: 1512.07496v2. doi: 10.1142/S0219891618500054.

[6]

D. Bresch and B. Desjardins, Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-233.  doi: 10.1007/s00220-003-0859-8.

[7]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pure. Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[8]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[9]

D. BreshB. Desjardins and E. Zatorska, Two-velosity hydrodynamics in fluid mechanics: Part Ⅱ Existence of global $\kappa$-entropy solutions to the compressible Navier-Stokes system with degenerate viscosities, J. Math. Pure Appl., 104 (2015), 801-836.  doi: 10.1016/j.matpur.2015.05.004.

[10]

Y. M. ChenS. J. Ding and B. L. Guo, Partial regularity for two-dimensional Landau-Lifshitz equations, Acta Math. Sinica, Eng. Ser., 14 (1998), 423-432.  doi: 10.1007/BF02580447.

[11]

S. J. Ding and C. Y. Wang, Finite time singularity of Landau-Lifshitz-Gilbert equations, Int. Math. Res. Notices, 2007 (2007), Art. ID rnm012, 25 pp. doi: 10.1093/imrn/rnm012.

[12]

J. W. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlinear Analysis: TMA, 73 (2010), 854-856.  doi: 10.1016/j.na.2010.03.047.

[13]

J. S. FanH. J. Gao and B. L. Guo, Regularity critera for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37.  doi: 10.1016/j.jmaa.2009.07.047.

[14]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford lecture series in Mathematics and its applications, vol. 26. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2004.

[15]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Analysis: TMA, 128 (2015), 106-121.  doi: 10.1016/j.na.2015.07.006.

[16]

B. L. Guo and S. J. Ding, Landau-Lifshitz Equations, World Scientific, 2008. doi: 10.1142/9789812778765.

[17]

B. L. Guo and M. C. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Cal. Var. Partial Diff. Eqns., 1 (1993), 311-334.  doi: 10.1007/BF01191298.

[18]

B. L. Guo and G. W. Wang, Global finite energy weak solution to the viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell equation in 2-dimension, Annl. Appl. Math., 32 (2016), 111-132. 

[19]

B. Haspot, Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D, Math. Nachr., 291 (2018), 2188-2203, arXiv: 1411.5503. doi: 10.1002/mana.201700050.

[20]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181.  doi: 10.2307/2000785.

[21]

D. Hoff, Global well-posedness of the cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Diff. Eqns., 95 (1992), 33-74.  doi: 10.1016/0022-0396(92)90042-L.

[22]

D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states, Z. Ange. Math. Phys., 49 (1998), 774-785.  doi: 10.1007/PL00001488.

[23]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anlaysis: RWA, 12 (2011), 1733-1735.  doi: 10.1016/j.nonrwa.2010.11.005.

[24]

A. Jüngel, Global weak solution to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[25]

Z. LeiD. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flow of harmonic maps in two dimensions, Proceedings of American Mathematical Society, 142 (2012), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0.

[26]

J. Li and Z. P. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826v2, 2015.

[27]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, vol.10. Oxford science publications, the Clarendon Press, Oxford University Press, New York, 1998.

[28]

X. G. Liu, Partial regularity for the Landau-Lifshitz system, Cal. Var. Partial Diff. Eqns., 20 (2004), 153-173.  doi: 10.1007/s00526-003-0231-z.

[29]

A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2007), 1344-1365.  doi: 10.1137/060658199.

[30]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.  doi: 10.1080/03605300600857079.

[31]

R. Moser, Partial regularity for the Landau-Lifshitz equation in small dimensions, MPI (Leipzig) preprint, 2002.

[32]

K. Nakamura and T. Sasada, Soliton and wave trains in ferromagnets, Phys. Lett. A, 48 (1974), 321-322.  doi: 10.1016/0375-9601(74)90447-2.

[33]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[34]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[35]

J. Simon, Compact sets in the space $L^p([0, T];B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[36]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[37]

A. F. Vasseur and C. Yu, Global weak solutions to the compressible quantum Navier-Stokes equations with damping, SIAM J. Math. Anal., 48 (2016), 1489-1511.  doi: 10.1137/15M1013730.

[38]

C. Y. Wang, On Landau-Lifshitz equation in dimensions at most four, Indiana University Mathematics Journal, 55 (2006), 1615-1644.  doi: 10.1512/iumj.2006.55.2810.

[39]

G. W. Wang and B. L. Guo, Existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz model in 2-dimension, Acta Mathematica Scientia, 37 (2017), 1361-1372.  doi: 10.1016/S0252-9602(17)30078-4.

[40]

E. Zatorska, On the flow of chemically reacting gaseous mixture, J. Diff. Eqns., 253 (2012), 3471-3500.  doi: 10.1016/j.jde.2012.08.043.

[41]

Y. L. ZhouB. L. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Sci. China Ser. A, 34 (1991), 257-266. 

[42]

Y. L. ZhouH. S. Sun and B. L. Guo, Existence of weak solution for boundary problems of systems of ferromagnetic chain, Sci. Sin. A, 27 (1981), 799-811. 

[43]

Y. L. ZhouH. S. Sun and B. L. Guo, On the solvability of the initial value problem for the quasilinear degenerate parabolic system: $\vec{Z}_t=\vec{Z}\times \vec{Z}_xx+\vec{f}(x, t, \vec{Z})$, Proc. Symp., 3 (1982), 713-732. 

[44]

Y. L. ZhouH. S. Sun and B. L. Guo, Finite difference solutions of the boundary problems for systems of ferromagnetic chain, J. Comp. Math., 1 (1983), 294-302. 

[45]

Y. L. ZhouH. S. Sun and B. L. Guo, Existence of weak solution for boundary problems of ferromagnetic chain, Sci. Sin. A, 27 (1984), 799-811. 

[46]

Y. L. ZhouH. S. Sun and B. L. Guo, The weak solution of homogeneous boundary value problem for the system of ferromagnetic chain with several variables, Sci. Sin. A, 4 (1986), 337-349. 

[47]

Y. L. ZhouH. S. Sun and B. L. Guo, Some boundary problems of the spin system and the system of ferro magnetic chain Ⅰ: Nonlinear boundary problems, Acta Math. Sci., 6 (1986), 321-337.  doi: 10.1016/S0252-9602(18)30514-9.

[48]

Y. L. ZhouH. S. Sun and B. L. Guo, Some boundary problems of the spin system and the system of ferromagnetic chain Ⅱ: Mixed problems and others, Acta Math. Sci., 7 (1987), 121-132.  doi: 10.1016/S0252-9602(18)30436-3.

[49]

Y. L. ZhouH. S. Sun and B. L. Guo, Weak solution systems of ferromagnetic chain with several variables, Science in China A, 30 (1987), 1251-1266. 

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces, (Second edition), Academic Press, Amsterdam, 2003.

[2]

A. I. Akhiezer, V. G. Yakhtar and S. V. Peletminskii, Spin Waves, North-Holland, 1968.

[3]

F. Alouges and A. Soyeur, On global weak solutions for Landau-Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal. TMA, 18 (1992), 1071-1084.  doi: 10.1016/0362-546X(92)90196-L.

[4]

P. Antonelli and S. Spirito, Global existence of finite energy weak solutions of quantum Navier-Stokes equations, Arch. Rational Mech. Anal., 225 (2017), 1161-1199.  doi: 10.1007/s00205-017-1124-1.

[5]

P. Antonelli and S. Spirito, On the compactness of finite energy weak solutions to the quantum Navier-Stokes equations, J. Hyperbolic Differ. Equ., 15 (2018), 133-147, arXiv: 1512.07496v2. doi: 10.1142/S0219891618500054.

[6]

D. Bresch and B. Desjardins, Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-233.  doi: 10.1007/s00220-003-0859-8.

[7]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pure. Appl., 86 (2006), 362-368.  doi: 10.1016/j.matpur.2006.06.005.

[8]

D. BreschB. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.  doi: 10.1081/PDE-120020499.

[9]

D. BreshB. Desjardins and E. Zatorska, Two-velosity hydrodynamics in fluid mechanics: Part Ⅱ Existence of global $\kappa$-entropy solutions to the compressible Navier-Stokes system with degenerate viscosities, J. Math. Pure Appl., 104 (2015), 801-836.  doi: 10.1016/j.matpur.2015.05.004.

[10]

Y. M. ChenS. J. Ding and B. L. Guo, Partial regularity for two-dimensional Landau-Lifshitz equations, Acta Math. Sinica, Eng. Ser., 14 (1998), 423-432.  doi: 10.1007/BF02580447.

[11]

S. J. Ding and C. Y. Wang, Finite time singularity of Landau-Lifshitz-Gilbert equations, Int. Math. Res. Notices, 2007 (2007), Art. ID rnm012, 25 pp. doi: 10.1093/imrn/rnm012.

[12]

J. W. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlinear Analysis: TMA, 73 (2010), 854-856.  doi: 10.1016/j.na.2010.03.047.

[13]

J. S. FanH. J. Gao and B. L. Guo, Regularity critera for the Navier-Stokes-Landau-Lifshitz system, J. Math. Anal. Appl., 363 (2010), 29-37.  doi: 10.1016/j.jmaa.2009.07.047.

[14]

E. Feireisl, Dynamics of Viscous Compressible Fluid, Oxford lecture series in Mathematics and its applications, vol. 26. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 2004.

[15]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Analysis: TMA, 128 (2015), 106-121.  doi: 10.1016/j.na.2015.07.006.

[16]

B. L. Guo and S. J. Ding, Landau-Lifshitz Equations, World Scientific, 2008. doi: 10.1142/9789812778765.

[17]

B. L. Guo and M. C. Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Cal. Var. Partial Diff. Eqns., 1 (1993), 311-334.  doi: 10.1007/BF01191298.

[18]

B. L. Guo and G. W. Wang, Global finite energy weak solution to the viscous quantum Navier-Stokes-Landau-Lifshitz-Maxwell equation in 2-dimension, Annl. Appl. Math., 32 (2016), 111-132. 

[19]

B. Haspot, Existence of global strong solution for the compressible Navier-Stokes equations with degenerate viscosity coefficients in 1D, Math. Nachr., 291 (2018), 2188-2203, arXiv: 1411.5503. doi: 10.1002/mana.201700050.

[20]

D. Hoff, Global existence for 1D, compressible, isentropic Navier-Stokes equations with large initial data, Trans. Amer. Math. Soc., 303 (1987), 169-181.  doi: 10.2307/2000785.

[21]

D. Hoff, Global well-posedness of the cauchy problem for the Navier-Stokes equations of nonisentropic flow with discontinuous initial data, J. Diff. Eqns., 95 (1992), 33-74.  doi: 10.1016/0022-0396(92)90042-L.

[22]

D. Hoff, Global solutions of the equations of one-dimensional, compressible flow with large data and forces, and with differing end states, Z. Ange. Math. Phys., 49 (1998), 774-785.  doi: 10.1007/PL00001488.

[23]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anlaysis: RWA, 12 (2011), 1733-1735.  doi: 10.1016/j.nonrwa.2010.11.005.

[24]

A. Jüngel, Global weak solution to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045.  doi: 10.1137/090776068.

[25]

Z. LeiD. Li and X. Y. Zhang, Remarks of global wellposedness of liquid crystal flows and heat flow of harmonic maps in two dimensions, Proceedings of American Mathematical Society, 142 (2012), 3801-3810.  doi: 10.1090/S0002-9939-2014-12057-0.

[26]

J. Li and Z. P. Xin, Global existence of weak solutions to the barotropic compressible Navier-Stokes flows with degenerate viscosities, arXiv: 1504.06826v2, 2015.

[27]

P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, vol.10. Oxford science publications, the Clarendon Press, Oxford University Press, New York, 1998.

[28]

X. G. Liu, Partial regularity for the Landau-Lifshitz system, Cal. Var. Partial Diff. Eqns., 20 (2004), 153-173.  doi: 10.1007/s00526-003-0231-z.

[29]

A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2007), 1344-1365.  doi: 10.1137/060658199.

[30]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.  doi: 10.1080/03605300600857079.

[31]

R. Moser, Partial regularity for the Landau-Lifshitz equation in small dimensions, MPI (Leipzig) preprint, 2002.

[32]

K. Nakamura and T. Sasada, Soliton and wave trains in ferromagnets, Phys. Lett. A, 48 (1974), 321-322.  doi: 10.1016/0375-9601(74)90447-2.

[33]

A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.

[34]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162. 

[35]

J. Simon, Compact sets in the space $L^p([0, T];B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[36]

A. F. Vasseur and C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations, Invent. Math., 206 (2016), 935-974.  doi: 10.1007/s00222-016-0666-4.

[37]

A. F. Vasseur and C. Yu, Global weak solutions to the compressible quantum Navier-Stokes equations with damping, SIAM J. Math. Anal., 48 (2016), 1489-1511.  doi: 10.1137/15M1013730.

[38]

C. Y. Wang, On Landau-Lifshitz equation in dimensions at most four, Indiana University Mathematics Journal, 55 (2006), 1615-1644.  doi: 10.1512/iumj.2006.55.2810.

[39]

G. W. Wang and B. L. Guo, Existence and uniqueness of the weak solution to the incompressible Navier-Stokes-Landau-Lifshitz model in 2-dimension, Acta Mathematica Scientia, 37 (2017), 1361-1372.  doi: 10.1016/S0252-9602(17)30078-4.

[40]

E. Zatorska, On the flow of chemically reacting gaseous mixture, J. Diff. Eqns., 253 (2012), 3471-3500.  doi: 10.1016/j.jde.2012.08.043.

[41]

Y. L. ZhouB. L. Guo and S. B. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Sci. China Ser. A, 34 (1991), 257-266. 

[42]

Y. L. ZhouH. S. Sun and B. L. Guo, Existence of weak solution for boundary problems of systems of ferromagnetic chain, Sci. Sin. A, 27 (1981), 799-811. 

[43]

Y. L. ZhouH. S. Sun and B. L. Guo, On the solvability of the initial value problem for the quasilinear degenerate parabolic system: $\vec{Z}_t=\vec{Z}\times \vec{Z}_xx+\vec{f}(x, t, \vec{Z})$, Proc. Symp., 3 (1982), 713-732. 

[44]

Y. L. ZhouH. S. Sun and B. L. Guo, Finite difference solutions of the boundary problems for systems of ferromagnetic chain, J. Comp. Math., 1 (1983), 294-302. 

[45]

Y. L. ZhouH. S. Sun and B. L. Guo, Existence of weak solution for boundary problems of ferromagnetic chain, Sci. Sin. A, 27 (1984), 799-811. 

[46]

Y. L. ZhouH. S. Sun and B. L. Guo, The weak solution of homogeneous boundary value problem for the system of ferromagnetic chain with several variables, Sci. Sin. A, 4 (1986), 337-349. 

[47]

Y. L. ZhouH. S. Sun and B. L. Guo, Some boundary problems of the spin system and the system of ferro magnetic chain Ⅰ: Nonlinear boundary problems, Acta Math. Sci., 6 (1986), 321-337.  doi: 10.1016/S0252-9602(18)30514-9.

[48]

Y. L. ZhouH. S. Sun and B. L. Guo, Some boundary problems of the spin system and the system of ferromagnetic chain Ⅱ: Mixed problems and others, Acta Math. Sci., 7 (1987), 121-132.  doi: 10.1016/S0252-9602(18)30436-3.

[49]

Y. L. ZhouH. S. Sun and B. L. Guo, Weak solution systems of ferromagnetic chain with several variables, Science in China A, 30 (1987), 1251-1266. 

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