2014, 13(4): 1481-1490. doi: 10.3934/cpaa.2014.13.1481

Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum

1. 

Department of Applied Mathematics, Nanjing Forestry University, Nanjing, 210037

2. 

Department of Mathematics, Nanjing University, Nanjing 210093

3. 

Department of Mathematics, Inha University, Incheon 402-751, South Korea

Received  June 2013 Revised  January 2014 Published  February 2014

In this paper we establish the global existence of strong solution to the density-dependent incompressible magnetohydrodynamic equations with vaccum in a bounded domain in $R^2$. Furthermore, the limit as the heat conductivity coefficient tends to zero is also obtained.
Citation: Jishan Fan, Fucai Li, Gen Nakamura. Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1481-1490. doi: 10.3934/cpaa.2014.13.1481
References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces,, 2nd ed., (2003).

[2]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,, Studies in Mathematics and its Applications, (1990).

[3]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^p$ theory,, \emph{J. Math. Fluid Mech.}, 12 (2010), 397. doi: 10.1007/2Fs00021-009-0295-4.

[4]

J.-S. Fan and F.-C. Li, Uniform local well-posedness to the density-dependent Navier-Stokes-Maxwell system,, \emph{Acta Appl. Math.}, (). doi: 10.1007/s10440-013-9857-9.

[5]

J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals,, Numerical Mathematics and Scientific Computation, (2006).

[6]

S. Itoh, On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incompressible fluid,, \emph{Glasgow Math. J.}, 36 (1994), 123. doi: 10.1017/S0017089500030639.

[7]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system,, \emph{J. Differential Equations}, 254 (2013), 511. doi: 10.1016/j.jde.2012.08.02.

[8]

M. L. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations,, \emph{Arch. Ration. Mech. Anal.}, 199 (2011), 739. doi: 10.1007/2Fs00205-010-0357-z.

[9]

T. Li and T. Qin, Physics and Partial Differential Equations,, Volume 1. Translated from the Chinese original by Yachun Li. Society for Industrial and Applied Mathematics (SIAM), (2012).

[10]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, The Clarendon Press, (1996).

[11]

E.-H. Lieb and M. Loss, Analysis,, 2nd ed., (2001).

[12]

A. Lunardi, Interpolation Theory,, 2nd ed., (2009).

[13]

T. Ozawa, On critical cases of Sobolev's inequalities,, \emph{J. Funct. Anal.}, 127 (1995), 259. doi: pii/S0022123685710129.

[14]

H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum,, \emph{Comput. Math. Appl.}, 61 (2011), 2742. doi: 10.1016/j.camwa.2011.03.03.

show all references

References:
[1]

R. A. Adams and J. F. Fournier, Sobolev Spaces,, 2nd ed., (2003).

[2]

S. N. Antontsev, A. V. Kazhikhov and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids,, Studies in Mathematics and its Applications, (1990).

[3]

H. Beirão da Veiga and F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An $L^p$ theory,, \emph{J. Math. Fluid Mech.}, 12 (2010), 397. doi: 10.1007/2Fs00021-009-0295-4.

[4]

J.-S. Fan and F.-C. Li, Uniform local well-posedness to the density-dependent Navier-Stokes-Maxwell system,, \emph{Acta Appl. Math.}, (). doi: 10.1007/s10440-013-9857-9.

[5]

J.-F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Methods for the Magnetohydrodynamics of Liquid Metals,, Numerical Mathematics and Scientific Computation, (2006).

[6]

S. Itoh, On the vanishing viscosity in the Cauchy problem for the equations of a nonhomogeneous incompressible fluid,, \emph{Glasgow Math. J.}, 36 (1994), 123. doi: 10.1017/S0017089500030639.

[7]

X. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system,, \emph{J. Differential Equations}, 254 (2013), 511. doi: 10.1016/j.jde.2012.08.02.

[8]

M. L. Lai, R. Pan and K. Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations,, \emph{Arch. Ration. Mech. Anal.}, 199 (2011), 739. doi: 10.1007/2Fs00205-010-0357-z.

[9]

T. Li and T. Qin, Physics and Partial Differential Equations,, Volume 1. Translated from the Chinese original by Yachun Li. Society for Industrial and Applied Mathematics (SIAM), (2012).

[10]

P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models,, The Clarendon Press, (1996).

[11]

E.-H. Lieb and M. Loss, Analysis,, 2nd ed., (2001).

[12]

A. Lunardi, Interpolation Theory,, 2nd ed., (2009).

[13]

T. Ozawa, On critical cases of Sobolev's inequalities,, \emph{J. Funct. Anal.}, 127 (1995), 259. doi: pii/S0022123685710129.

[14]

H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum,, \emph{Comput. Math. Appl.}, 61 (2011), 2742. doi: 10.1016/j.camwa.2011.03.03.

[1]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207

[2]

Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359

[3]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041

[4]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

[5]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395

[6]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400

[7]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

[8]

Jacques A. L. Silva, Flávia T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5 (4) : 843-857. doi: 10.3934/mbe.2008.5.843

[9]

Qing Chen, Zhong Tan. Global existence in critical spaces for the compressible magnetohydrodynamic equations. Kinetic & Related Models, 2012, 5 (4) : 743-767. doi: 10.3934/krm.2012.5.743

[10]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373

[11]

J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054

[12]

Baojun Song, Wen Du, Jie Lou. Different types of backward bifurcations due to density-dependent treatments. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1651-1668. doi: 10.3934/mbe.2013.10.1651

[13]

Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008

[14]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

[15]

Zhong Tan, Qiuju Xu, Huaqiao Wang. Global existence and convergence rates for the compressible magnetohydrodynamic equations without heat conductivity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5083-5105. doi: 10.3934/dcds.2015.35.5083

[16]

Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763

[17]

Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157

[18]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-21. doi: 10.3934/dcdsb.2017209

[19]

Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure & Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647

[20]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]