This paper is concerned with a stability problem on perturbations near a physically important steady state solution of the 3D MHD system. We obtain three major results. The first assesses the existence of global solutions with small initial data. Second, we derive the temporal decay estimate of the solution in the L2-norm, where to prove the result, we need to overcome the difficulty caused by the presence of linear terms from perturbation. Finally, the decay rate in L2 space for higher order derivatives of the solution is established.
Xueli KE
,
Baoquan YUAN
,
Yaomin XIAO
. A STABILITY PROBLEM FOR THE 3D MAGNETOHYDRODYNAMIC EQUATIONS NEAR EQUILIBRIUM[J]. Acta mathematica scientia, Series B, 2021
, 41(4)
: 1107
-1118
.
DOI: 10.1007/s10473-021-0405-9
[1] Abidi H, Zhang P. On the global solution for a 3-D MHD system with initial data near equilibrium. Comm Pure Appl Math, 2017, 70(8):1509-1561
[2] Alfvén H. Existence of electromagnetic-hydrodynamic waves. Nature, 1942, 150(2):3763-3767
[3] Biskamp D. Nonlinear Magnetohydrodynamics. Cambridge:Cambridge University Press, 1993
[4] Cao C S, Wu J H, Yuan B Q. The 2D incompressible magnetohydrodynamics equations with only magnetic diffusion. SIAM J Math Anal, 2013, 46(1):588-602
[5] Davidson P. An Introduction to Magnetohydrodynamics. Cambridge:Cambridge University Press, 2001
[6] Deng W, Zhang P. Large time behavior of solutions to 3-D MHD system with initial data near equilibrium. Arch Ration Mech Anal, 2018, 230(3):1017-1102
[7] Dong B Q, Li J N, Wu J H. Global well-posedness and large-time decay for the 2D micropolar equations. J Differential Equations, 2017, 262(6):3488-3523
[8] Duvaut G, Lions J L. Inéquations en thermoélasticité et magnétohydrodynamique. Arch Ration Mech Anal, 1972, 46(4):241-279
[9] He C, Xin Z P. On the regularity of weak solutions to the magnetohydrodynamic equations. J Differential Equations, 2005, 213(2):235-254
[10] Ji R H, Lin H X, Wu J H, et al. Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation. Appl Math Lett, 2019, 94:244-249
[11] Jiu Q S, He C. Remarks on the regularity to 3-D ideal magnetohydrodynamic equations. Acta Math Sin, 2004, 20(4):695-708
[12] Jiu Q S, Yu H. Decay of solutions to the three-dimensional generalized Navier-Stokes equations. Asymptot Anal, 2015, 94(1/2):105-124
[13] Kato T, Ponce G. Commutator estimates and the Euler and Navier-Stokes equations. Comm Pure Appl Math, 1988, 41(7):891-907
[14] Kenig C, Ponce G, Vega L. Well-posedness of the initial value problem for the Korteweg-Vries equation. J Amer Math Soc, 1991, 4(2):323-347
[15] Li M, Shang H F. Large time decay of solutions for the 3D magneto-micropolar equations. Nonlinear Anal Real World Appl, 2018, 44:479-496
[16] Lin F H, Xu L, Zhang P. Global small solutions of 2-D incompressible MHD system. J Differential Equations,
[19] Priest E. Magnetic Reconnection:MHD Theory and Applications. Cambridge:Cambridge University Press, 2000
[20] Ren X X, Wu J H, Xiang Z Y, et al. Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion. J Funct Anal, 2014, 267(2):503-541
[21] Ren X X, Xiang Z Y, Zhang Z F. Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion. Sci China Math, 2016, 59(10):1949-1974
[22] Schonbek M E, Large time behavior of solutions to the Navier-Stokes equations. Comm Partial Differential Equations, 1983, 11(7):733-763
[23] Schonbek M E, Dafermos C. L2 decay for weak solutions of the Navier-Stokes equations. Arch Rational Mech Anal, 1985, 88(3):209-222
[24] Schonbek M E, Schonbek T P, Süli Endre. Large-time behaviour of solutions to the magnetohydrodynamics equations. Math Ann, 1996, 304(1):717-756
[25] Sermange M, Temam R. Some mathematical questions related to the MHD equations. Comm Pure Appl Math, 1983, 36(5):635-664
[26] Wan R H. Optimal decay estimate of strong solutions for the 3D incompressible Oldroyd-B model without damping. Pacific J Math, 2019, 301(2):667-701
[27] Wei D Y, Zhang Z F. Global well-posedness of the MHD equations in a homogeneous magnetic field. Anal PED, 2017, 10(6):1361-1406
[28] Wu J H, Zhu Y. Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium. Adv Math, 2021, 377(1):107466
[29] Ye Z. Global regularity of the two-dimensional regularized MHD equations. Dyn Partial Differ Equ, 2019, 16(2):185-223
[30] Yuan B Q, Liu Y. Global existence and decay rate of strong solution to incompressible Oldroyd type model equations. Rocky Mountain J Math, 2018, 48(5):1703-17202015, 259(10):5440-5485
[17] Majda A, Bertozzi A. Vorticity and Incompressible Flow. Cambridge:Cambridge University Press, 2002
[18] Miao C X, Yuan B Q, Zhang B. Well-posedness for the incompressible magnetohydrodynamic system. Math Methods Appl Sci, 2010, 30(8):961-976
