The main objective of this paper is to study the global strong solution of the parabolic-hyperbolic incompressible magnetohydrodynamic model in the two dimensional space. Based on Agmon, Douglis, and Nirenberg’s estimates for the stationary Stokes equation and Solonnikov’s theorem on Lp-Lq-estimates for the evolution Stokes equation, it is shown that this coupled magnetohydrodynamic equations possesses a global strong solution. In addition, the uniqueness of the global strong solution is obtained.
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S. Agmon, A. Douglis, L. Nirenberg: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Commun. Pure Appl. Math. 17 (1964), 35–92.
C. Amrouche, V. Girault: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech. Math. J. 44 (1994), 109–140.
C. Cao, J. Wu: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226 (2011), 1803–1822.
S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics, Clarendon Press, Oxford, 1961.
G. Duvaut, J. L. Lions: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46 (1972), 241–279. (In French.)
M. Geissert, M. Hess, M. Hieber, C. Schwarz, K. Stavrakidis: Maximal Lp-Lq-estimates for the Stokes equation: a short proof of Solonnikov’s theorem. J. Math. Fluid Mech. 12 (2010), 47–60.
C. He, Z. Xin: On the regularity of weak solutions to the magnetohydrodynamic equations. J. Differ. Equations 213 (2005), 235–254.
X. Huang, Y. Wang: Global strong solution to the 2D nonhomogeneous incompressible MHD system. J. Differ. Equations 254 (2013), 511–527.
Q. Jiu, D. Niu, J. Wu, X. Xu, H. Yu: The 2D magnetohydrodynamic equations with magnetic diffusion. Nonlinearity 28 (2015), 3935–3955.
R. Liu, Q. Wang: S1 attractor bifurcation analysis for an electrically conducting fluid flow between two rotating cylinders. Physica D 392 (2019), 17–33.
R. Liu, J. Yang: Magneto-hydrodynamical model for plasma. Z. Angew. Math. Phys. 68 (2017), Article number 114, 15 pages.
T. Ma: Theory and Method of Partial Differential Equation. Science Press, Beijing, 2011. (In Chinese.)
T. Ma, S. Wang: Phase Transition Dynamics. Springer, New York, 2014.
N. Masmoudi: Global well posedness for the Maxwell-Navier-Stokes system in 2D. J. Math. Pures Appl. (9) 93 (2010), 559–571.
D. Regmi: Global weak solutions for the two-dimensional magnetohydrodynamic equations with partial dissipation and diffusion. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 144 (2016), 157–164.
X. Ren, J. Wu, Z. Xiang, Z. Zhang: Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J. Funct. Anal. 267 (2014), 503–541.
T. Sengul, S. Wang: Pattern formation and dynamic transition for magnetohydrodynamic convection.
M. Sermange, R. Temam: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36 (1983), 635–664.
V. A. Solonnikov: Estimates for solutions of nonstationary Navier-Stokes equations. J. Sov. Math. 8 (1977), 467–529.
R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. American Mathematical Society, Providence, 2001.
V. Vialov: On the regularity of weak solutions to the MHD system near the boundary. J. Math. Fluid Mech. 16 (2014), 745–769.
Q. Wang: Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete Contin. Dyn. Syst., Ser. B. 19 (2014), 543–563.
T. Wang: A regularity criterion of strong solutions to the 2D compressible magnetohy-drodynamic equations. Nonlinear Anal., Real World Appl. 31 (2016), 100–118.
J. Wu: Generalized MHD equations. J. Differ. Equations 195 (2003), 284–312.
K. Yamazaki: Global regularity of N-dimensional generalized MHD system with anisotropic dissipation and diffusion. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 122 (2015), 176–191.
X. Zhong: Global strong solutions for 3D viscous incompressible heat conducting magnetohydrodynamic flows with non-negative density. J. Math. Anal. Appl. 446 (2017), 707–729.
The work is supported by the Young Scholars Development Fund of SWPU (Grant No. 201899010079), the scientific research starting project of SWPU (Grant No. 2018QHZ029), the Natural Science Foundation of China (11771306), the National Nature Science Foundation of China (Key Program) (No. 51534006), Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and Its Applications (No. 18TD0013), Southwest Petroleum Univ. under Grant 2019CXTD08, Youth Science and Technology Innovation Team of Southwest Petroleum Univ. for Nonlinear Systems (No. 2017CXTD02), the Youth of Natural Science of SWMU (No. 2019ZQN146).
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Liu, R., Yang, J. Global Strong Solutions of a 2-D New Magnetohydrodynamic System. Appl Math 65, 105–120 (2020). https://doi.org/10.21136/AM.2020.0208-19
- global strong solution
- Stokes equation