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  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 & 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. Google Scholar

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[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. Google Scholar

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[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. Google Scholar

[34]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[2]

Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199

[3]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[4]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215

[5]

Wei Deng, Baisheng Yan. On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics. Evolution Equations & Control Theory, 2013, 2 (4) : 599-620. doi: 10.3934/eect.2013.2.599

[6]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[7]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[8]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279

[9]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

[10]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161

[11]

Peter Anthony, Sergey Zelik. Infinite-energy solutions for the Navier-Stokes equations in a strip revisited. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1361-1393. doi: 10.3934/cpaa.2014.13.1361

[12]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371

[13]

Fei Jiang, Song Jiang, Junpin Yin. Global weak solutions to the two-dimensional Navier-Stokes equations of compressible heat-conducting flows with symmetric data and forces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 567-587. doi: 10.3934/dcds.2014.34.567

[14]

Joanna Rencławowicz, Wojciech M. Zajączkowski. Global regular solutions to the Navier-Stokes equations with large flux. Conference Publications, 2011, 2011 (Special) : 1234-1243. doi: 10.3934/proc.2011.2011.1234

[15]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085

[16]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

[17]

Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284

[18]

Ciprian Foias, Ricardo Rosa, Roger Temam. Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1611-1631. doi: 10.3934/dcds.2010.27.1611

[19]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[20]

Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088

2018 Impact Factor: 1.008

Article outline

[Back to Top]