# American Institute of Mathematical Sciences

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

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

 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).

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

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##### 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
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