# American Institute of Mathematical Sciences

## Global strong solution and exponential decay for nonhomogeneous magnetohydrodynamic equations

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

Received  March 2020 Revised  May 2020 Published  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 magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 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] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [2] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354 [3] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273 [4] Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105 [5] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [6] Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [7] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [8] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [9] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [10] Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299 [11] Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026 [12] Martin Kalousek, Joshua Kortum, Anja Schlömerkemper. Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 17-39. doi: 10.3934/dcdss.2020331 [13] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [14] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [15] Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020386 [16] Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119 [17] Haiyu Liu, Rongmin Zhu, Yuxian Geng. Gorenstein global dimensions relative to balanced pairs. Electronic Research Archive, 2020, 28 (4) : 1563-1571. doi: 10.3934/era.2020082 [18] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [19] Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344 [20] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

2019 Impact Factor: 1.27