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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 and 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[23]

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

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[23]

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

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