LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS

doi:10.1007/s10473-020-0210-x

Abstract

Abstract: This article is concerned with the 3D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain. We obtain weighted estimates of the velocity and magnetic field, and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.

Key words: Nonhomogeneous MHD equations, local existence and uniqueness, vacuum, t-weighted H² estimate, Galerkin approximation

CLC Number:

35Q35

Hongmin LI, Yuelong XIAO. LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS[J].Acta mathematica scientia,Series B, 2020, 40(2): 442-456.

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References

[1] Agapito R, Schonbek M. Non-uniform decay of MHD equations with and without magnetic diffusion. Commu Part Diff Eq, 2007, 32(11):1791-1821
[2] Beirão da Veiga H, Valli A. On the Euler equations for nonhomogeneous fluids II. J Math Anal Appl, 1980, 73(2):338-350
[3] Berselli L, Spirito S. On the vanishing viscosity limit of 3D Navier-Stokes equations under slip boundary conditions in general domains. Commun Math Phys, 2012, 316(1):171-198
[4] Biskamp D. Nonlinear magnetohydrodynamics. Cambridge University Press, 1997
[5] Chen G Q, Wang D H. Existence and continuous dependence of large solutions for the magnetohydrodynamic equations. Z Angew Math Phys, 2003, 54(4):608-632
[6] Chen P F, Xiao Y L, Zhang H. Vanishing viscosity limit for the 3D nonhomogeneous incompressible NavierStokes equation with special slip boundary conditions. Electron J Differ Eq, 2017, 2017(169):1-13
[7] Chen Q, Tan Z, Wang Y J. Strong solutions to the incompressible magnetohydrodynamic equations. Math Method Appl Sci, 2011, 34(1):94-107
[8] Ducomet B, Feireisl E. The equations of Magnetohydrodynamics:on the interaction between matter and radiation in the evolution of gaseous stars. Commun Math Phys, 2006, 266(3):595-629
[9] Fan J S, Jiang S, Nakamura G. Vanishing shear viscosity limit in the magnetohydrodynamic equations. Commun Math Phys, 2007, 270(3):691-708
[10] Feireisl E, Novotný A, Petzeltová H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3(4):358-392
[11] Ferreira L C F, Planas G, Vllamizar-Roa E J. On the Nonhomogeneous Navier-Stokes System with Navier Friction Boundary Conditions. SIAM J Math Anal, 2013, 45(4):2576-2595
[12] Freidberg J. Ideal Magnetohydrodynamics. New York London Plenum Press, 1987
[13] Guo B L, Wang G W. Vanishing viscosity limit for the 3D magnetohydrodynamic system with generalized Navier slip boundary conditions. Math Meth Appl Sci, 2016, 39(15):4526-4534
[14] Hoff D, Tsyganov E. Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics. Z Angew Math Phys, 2005, 56(5):791-804
[15] Hu X P, Wang D H. Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch Rational Mech Anal, 2010, 197(1):203-238
[16] Huang X D, Wang Y. Global strong solution to the 2D nonhomogeneous incompressible MHD system. J Differ Equations, 2013, 254(2):511-527
[17] Huang X D, Wang Y. Global strong solution of 3D inhomogeneous Navier-Stokes equations with densitydependent viscosity. J Differ Equations, 2015, 259(4):1606-1627
[18] Jiu Q S, Wang Y, Xin Z P. Global well-posedness of the Cauchy problem of two-dimensional compressible Navier-Stokes equations in weighted spaces. J Differ Equations, 255(3):351-404
[19] Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Flow. second English edition, Mathematics and Its Applications. Vol. 2. New York, London, Paris:Gordon and Breach, Science Publishers, 1969
[20] Ladyzhenskaya O A, Solonnikov V A. Unique solvability of an initial and boundary value problem for viscous incompressible non-homogeneous fluids. J Sov Math, 1978, 9(5):697-749
[21] Li F C, Zhang Z P. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete and continuous dynamical systems, 2018, 38(9):4279-4304
[22] Li H Y. Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity. Math Methods Appl Sci, 2018, 41(8):3062-3092
[23] Li J. Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density. J Differ Equations, 2017, 263(10):6512-6536
[24] Lv B Q, Xu Z H, Zhong X. 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, 2017, 108(1):41-62
[25] Paicu M, Zhang P, Zhang Z F. Global unique solvability of homogeneous Navier-Stokes equations with bounded density. Commun Part Diff Eq, 2013, 38(7):1208-1234
[26] Sart R. Existence of finite energy weak solutions for the equations MHD of compressible fluids. Appl Anal, 2009, 88(3):357-379
[27] Simon J. Nonhomogeneous viscous incompressible fluids:existence of velocity, density, and pressure. SIAM J Math Anal, 1990, 21(5):1093-1117
[28] Temam R. Navier-Stokes equations and nonlinear functional analysis. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia:SIAM, 1983
[29] Wu H W. Strong solutions to the incompressible magnetohydrodynamic equations with vacuum. Comput Math Appl, 2011, 61(9):2742-2753
[30] Wu J H. Generalized MHD equations. J Differ Equations, 2003, 195(2):284-312
[31] Xiao Y L, Xin Z P. On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun Pur Appl Math, 2007, 60(7):1027-1055
[32] Xiao Y L, Xin Z P, Wu J H. Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition. J Funct Anal, 2009, 257(11):3375-3394
[33] Xu L, Zhang P. Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J Math Anal, 2015, 47(1):26-65
[34] Yu H B, Zhang P X, Shi X J. Global strong solutions to the 3D incompressible MHD equations with density-dependent viscosity. Comput Math Appl, 2018, 75(8):2825-2834