American Institute of Mathematical Sciences

Advanced
July  2021, 26(7): 3563-3578. doi: 10.3934/dcdsb.2020246

Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations

Xin Zhong

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  March 2020 Revised  May 2020 Published  July 2021 Early access  August 2020

Fund Project: Supported by National Natural Science Foundation of China (No. 11901474)

The present paper concerns an initial boundary value problem of two-dimensional (2D) nonhomogeneous magnetohydrodynamic (MHD) equations with non-negative density. We establish the global existence and exponential decay of strong solutions. In particular, the initial data can be arbitrarily large. The key idea is to use a lemma due to Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

Keywords: Nonhomogeneous magnetohydrodynamic equations, global strong solution, exponential decay, large initial data, vacuum.
Mathematics Subject Classification: 76W05;76D03.
Citation: Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246
References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.  Google Scholar

[2]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.   Google Scholar

[3]

Q. Bie, Q. Wang and Z. Yao, Global well-posedness of the 3D incompressible MHD equations with variable density, Nonlinear Anal. Real World Appl., 47 (2019), 85-105.  doi: 10.1016/j.nonrwa.2018.10.008.  Google Scholar

[4]

F. Chen, B. Guo and X. Zhai, Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.  doi: 10.3934/krm.2019002.  Google Scholar

[5]

F. Chen, Y. Li and H. Xu, Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data, Discrete Contin. Dyn. Syst., 36 (2016), 2945-2967.  doi: 10.3934/dcds.2016.36.2945.  Google Scholar

[6]

Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.  doi: 10.1002/mma.1338.  Google Scholar

[7]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.  Google Scholar

[8]

R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.  doi: 10.1002/cpa.21806.  Google Scholar

[9]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. Google Scholar

[12]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[13]

H. Li, Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.  doi: 10.1002/mma.4801.  Google Scholar

[14]

J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar

[15]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.  Google Scholar

[16] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.   Google Scholar
[17]

Y. Liu, Global existence and exponential decay of strong solutions for the 3D incompressible MHD equations with density-dependent viscosity coefficient, Z. Angew. Math. Phys., 70 (2019), Paper No. 107. doi: 10.1007/s00033-019-1157-4.  Google Scholar

[18]

B. Lü, X. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.  Google Scholar

[19]

B. Lü, Z. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.  Google Scholar

[20]

M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.  Google Scholar

[21]

X. Si and X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys., 67 (2016), Paper No. 126. doi: 10.1007/s00033-016-0722-3.  Google Scholar

[22]

S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), Paper No. 23. doi: 10.1007/s00033-018-0915-z.  Google Scholar

[23]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^th$ edition, Springer-Verlag, Berlin, 2008.  Google Scholar

show all references

References:
[1]

H. Abidi and M. Paicu, Global existence for the magnetohydrodynamic system in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 447-476.  doi: 10.1017/S0308210506001181.  Google Scholar

[2]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.   Google Scholar

[3]

Q. Bie, Q. Wang and Z. Yao, Global well-posedness of the 3D incompressible MHD equations with variable density, Nonlinear Anal. Real World Appl., 47 (2019), 85-105.  doi: 10.1016/j.nonrwa.2018.10.008.  Google Scholar

[4]

F. Chen, B. Guo and X. Zhai, Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density, Kinet. Relat. Models, 12 (2019), 37-58.  doi: 10.3934/krm.2019002.  Google Scholar

[5]

F. Chen, Y. Li and H. Xu, Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data, Discrete Contin. Dyn. Syst., 36 (2016), 2945-2967.  doi: 10.3934/dcds.2016.36.2945.  Google Scholar

[6]

Q. Chen, Z. Tan and Y. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.  doi: 10.1002/mma.1338.  Google Scholar

[7]

H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for nonhomogeneous incompressible fluids, Comm. Partial Differential Equations, 28 (2003), 1183-1201.  doi: 10.1081/PDE-120021191.  Google Scholar

[8]

R. Danchin and P. B. Mucha, The incompressible Navier-Stokes equations in vacuum, Comm. Pure Appl. Math., 72 (2019), 1351-1385.  doi: 10.1002/cpa.21806.  Google Scholar

[9]

B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., 137 (1997), 135-158.  doi: 10.1007/s002050050025.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

A. Friedman, Partial Differential Equations, Dover Books on Mathematics, New York, 2008. Google Scholar

[12]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.  doi: 10.1016/j.jde.2012.08.029.  Google Scholar

[13]

H. Li, Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.  doi: 10.1002/mma.4801.  Google Scholar

[14]

J. Li, Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density, J. Differential Equations, 263 (2017), 6512-6536.  doi: 10.1016/j.jde.2017.07.021.  Google Scholar

[15]

Z. Liang, Local strong solution and blow-up criterion for the 2D nonhomogeneous incompressible fluids, J. Differential Equations, 258 (2015), 2633-2654.  doi: 10.1016/j.jde.2014.12.015.  Google Scholar

[16] P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. Ⅰ: Incompressible Models, Oxford University Press, Oxford, 1996.   Google Scholar
[17]

Y. Liu, Global existence and exponential decay of strong solutions for the 3D incompressible MHD equations with density-dependent viscosity coefficient, Z. Angew. Math. Phys., 70 (2019), Paper No. 107. doi: 10.1007/s00033-019-1157-4.  Google Scholar

[18]

B. Lü, X. Shi and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent Navier-Stokes equations with vacuum, Nonlinearity, 31 (2018), 2617-2632.  doi: 10.1088/1361-6544/aab31f.  Google Scholar

[19]

B. Lü, Z. Xu and X. Zhong, Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum, J. Math. Pures Appl., 108 (2017), 41-62.  doi: 10.1016/j.matpur.2016.10.009.  Google Scholar

[20]

M. Paicu, P. Zhang and Z. Zhang, Global unique solvability of inhomogeneous Navier-Stokes equations with bounded density, Comm. Partial Differential Equations, 38 (2013), 1208-1234.  doi: 10.1080/03605302.2013.780079.  Google Scholar

[21]

X. Si and X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys., 67 (2016), Paper No. 126. doi: 10.1007/s00033-016-0722-3.  Google Scholar

[22]

S. Song, On local strong solutions to the three-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and vacuum, Z. Angew. Math. Phys., 69 (2018), Paper No. 23. doi: 10.1007/s00033-018-0915-z.  Google Scholar

[23]

M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4$^th$ edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[1]

Xin Zhong. Global strong solution to the nonhomogeneous micropolar fluid equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021296
[2]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
[3]

Fei Chen, Yongsheng Li, Huan Xu. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 2945-2967. doi: 10.3934/dcds.2016.36.2945
[4]

Xin Zhong. Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum. Communications on Pure & Applied Analysis, 2022, 21 (2) : 493-515. doi: 10.3934/cpaa.2021185
[5]

Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026
[6]

Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[7]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163
[8]

Xin Zhong. Global well-posedness to the cauchy problem of two-dimensional density-dependent boussinesq equations with large initial data and vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6713-6745. doi: 10.3934/dcds.2019292
[9]

Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481
[10]

Xiaoli Li. Global strong solution for the incompressible flow of liquid crystals with vacuum in dimension two. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4907-4922. doi: 10.3934/dcds.2017211
[11]

Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973
[12]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
[13]

Xin Zhong. A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4603-4615. doi: 10.3934/dcdsb.2020115
[14]

Qiang Tao, Ying Yang. Exponential stability for the compressible nematic liquid crystal flow with large initial data. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1661-1669. doi: 10.3934/cpaa.2016007
[15]

Xiaoping Zhai, Zhaoyang Yin. Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2829-2859. doi: 10.3934/dcds.2017122
[16]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016
[17]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[18]

Kunquan Li, Yaobin Ou. Global wellposedness of vacuum free boundary problem of isentropic compressible magnetohydrodynamic equations with axisymmetry. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 487-522. doi: 10.3934/dcdsb.2021052
[19]

Zefu Feng, Changjiang Zhu. Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3069-3097. doi: 10.3934/dcds.2019127
[20]

Matthieu Alfaro, Arnaud Ducrot. Sharp interface limit of the Fisher-KPP equation when initial data have slow exponential decay. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 15-29. doi: 10.3934/dcdsb.2011.16.15

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (260)
  • HTML views (311)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy