Articles

DECAY ESTIMATES FOR ISENTROPIC COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS IN BOUNDED DOMAIN

  • Mohamed Ahmed Abdallah ,
  • JIANG Fei ,
  • TAN Zhong
Expand
  • 1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
    2. College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350108, China;
    3. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Received date: 2011-06-13

  Revised date: 2011-12-03

  Online published: 2012-11-20

Supported by

Supported by the National Natural Science Foundation of China (10976026), and the Fujian Provincial Department of Science and Technology (JK2009045).

Abstract

In this paper, under the hypothesis that ρ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible magnetohydrody-namic equations and show that the weak solutions decay exponentially to the equilibrium state in L2 norm. Our result verifies that the method of Daoyuan Fang, Ruizhao Zi and Ting Zhang [1] can be adapted to magnetohydrodynamic equations.

Cite this article

Mohamed Ahmed Abdallah , JIANG Fei , TAN Zhong . DECAY ESTIMATES FOR ISENTROPIC COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS IN BOUNDED DOMAIN[J]. Acta mathematica scientia, Series B, 2012 , 32(6) : 2211 -2220 . DOI: 10.1016/S0252-9602(12)60171-4

References

[1] Fan D -Y, Zi R -Z, Zhang T. Decay estimates for isentropic compressible Navier–Stokes equations in bounded domain. J Math Anal Appl, 2012, 386: 939–947

[2] Cabannes H. Theoretical Magnetofluid-Dynamics. New York: Academic Press, 1970

[3] Kulikovskiy A, Lyubimov G. Magnetohydrodynamics. New York: Addison-Wesley, 1965

[4] Chandrasekhar S. An Introduction to the Study of Stellar Structures. Chicago: University of Chicago Press, 1938

[5] Lions P. Mathematical Topics in Fluid Mechanics: Compressible Models. Oxford: Oxford University Press, 1998

[6] DiPerna R, Lions P -L. Ordinary differential equations, transport theory and Sobolev spaces. Invent Math, 1989, 98: 511–547

[7] Feireisl E, Novotn`y A, Petzeltov´a H. On the existence of globally defined weak solutions to the Navier-Stokes equations. J Math Fluid Mech, 2001, 3: 358–392

[8] Hoff D. Spherically symmetric solutions of the Navier-Stokes equations for compressible, isothermal flow with large discontinuous initial data. Indiana Univ Math J, 1992, 41: 1225–1302

[9] Jiang S, Zhang P. Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data. Indiana Univ Math J, 2002, 51: 345–355

[10] Jiang S, Zhang P. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Commun Math Phys, 2001, 215: 559–581

[11] Jiang S, Zhang P. Axisymmetric solutions of the 3-D Navier-Stokes-equations for compressible isentropic fluids. J Math Pures Appl, 2003, 82: 949–973

[12] Hu X,Wang D -H. Global existence and large-Time behavior of solutions to the three-dimensional equations of cmpressible magnetohydrodynamic flows. Arch Rational Mech Anal, 2010, 197: 203–238

[13] Feireisl E, Petzeltov´a H. Large-time dehaviour of solutions to the Navier-Stokes equations of compressible flow. Arch Rational Mech Anal, 1999, 150: 77–96

[14] Jiang F, Tan Z, Wang H -Q. A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with coulomb force. J Math Anal Appl, 2011, 379: 316–324

[15] Duan R, Ukai S, Yang T, Zhao H -J. Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Math Models Methods Appl Sci, 2007, 17: 737–758

[16] Kagei Y, Kobayashi T. Asymptotic behavior of solutions of the compressible navier-stokes equations on the half space. Arch Rational Mech Anal, 2005, 177: 231–330

[17] Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3. Comm Math Phys, 1999, 200: 621–659

[18] Kagei Y, Kobayashi T. On large time behavior of solutions to the compressible navier-stokes equations in the half space in R3. Arch Rational Mech Anal, 2002, 165: 89–159

[19] Kobayashi T. Some estimates of solutions for the equations of motion of compressible viscous fluid in an exterior domain in R3. J Differ Equ, 2002, 184: 587–619

[20] Stra?skraba I, Zlotnik A A. On a decay rate for 1d-viscous compressible barotropic fluid equations. J Evol Equ, 2002, 2: 69–96

[21] Matsumura A, Nishida T. Initial-boundary value problems for the equations of motion of general fluids, computing methods in applied sciences and engineering. J Math Kyoto Univ, 1982, 20: 389–406

[22] Matsumura A, Nishida T. Initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids. Comm Math Phys, 1983, 89: 445–464

[23] Matsumura A, Nishida T. The initial value problem for the equation of compressible viscous and heatconductive fluids. Proc Jpn Acad Ser-A, 1979, 55: 337–342

[24] Matsumura A, Nishida T. The initial value problem for the equation of motion of viscous and heatconductive gases. J Math Kyoto Univ, 1980, 20: 67–104

[25] Ponce G. Global existence of small solution to a class of nonlinear evolution equations. Nonlinear Anal, 1985, 9: 339–418

[26] Chen Q, Tan Z. Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations. Nonlinear Analysis: TMA, 2010, 72: 4438–4451

[27] Novotn`y A, Straˇskraba I. Introduction to the Mathematical Theory of Compressible Flow. Oxford: Oxford University Press, 2004

[28] Jiang F, Tan Z. On radially symmetric solutions of the compressible isentropic self-gravitating fluid. Nonlinear Analysis: TMA, 2010, 72: 3463–3483

[29] Feireisl E, Novotn`y A, Petzeltov´a H. On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid. Math Meth Appl Sci, 2002, 25: 1045–1073

[30] Jiang F, Tan Z. On the domain dependence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow. Math Methods Appl Sci, 2009, 32(18): 2350–2367

[31] Galdi G. An Introduction to the Mathematical Theory of the Navier-Stokes Equations. New York: Springer-Verlag, 1994

[32] Feireisl E, Petzeltov´a H. On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flow. Commun Partial Differ Equ, 1999, 25: 755–767

[33] Feireisl E, Novotn`y A. Singular Limits in Thermodynamics of Viscous Fluids. Basel: Birkhäuser Verlag, 2009

[34] Wang W W, Jiang F, Gao Z S. Sequential stability of weak solutions in compressible self-gravitating fluids and stationary problem. Math Meth Appl Sci, 2012, 35(9): 1014–1032

[35] Barker B, Lafitte O, Zumbrun K. Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity. Acta Math Sci, 2010, 30B(2): 447–498

[36] Klingenberg C, Waagan K. Relaxation solvers for ideal MHD equations – a review. Acta Math Sci, 2010, 30B(2): 621–632

[37] Yuan B Q. Regularity of weak solutions to magneto-micropolar fluid equations. Acta Math Sci, 2010, 30B(5): 1469–1480

[38] Wang Y Z, Wang S B, Wang Y X. Regularity criteria for weak solution to the 3D magnetohydrodynamic equations. Acta Math Sci, 2012, 32B(3): 1063–1072

[39] Zhang J W. The inviscid and non-resistive limit in the cauchy problem for 3-D nonhomogeneous incompressible magneto-hydrodynamics. Acta Math Sci, 2011, 31B(3): 882–896

Outlines

/