三维可压缩 MHD 方程大解的时间衰减率

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1397-1408.

三维可压缩 MHD 方程大解的时间衰减率

陈菲^1,^*(),王帅¹(),赵永叶²(),王传宝¹()

¹青岛大学数学与统计学院山东青岛 266071
²广州航海学院广州 510725

收稿日期:2022-08-15 修回日期:2023-04-10 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 陈菲 E-mail:feichenstudy@163.com;shuai172021@163.com;yongyezhao@163.com;wcb1216@163.com
作者简介:王帅，Email: shuai172021@163.com;|赵永叶，Email: yongyezhao@163.com;|王传宝，Email: wcb1216@163.com
基金资助:
国家自然科学基金(12101345);山东省自然科学基金(ZR2021QA017);广州市基础研究计划基础与应用基础研究项目(202102020283)

Time Decay Rate for Large-Solution About 3D Compressible MHD Equations

Chen Fei^1,^*(),Wang Shuai¹(),Zhao Yongye²(),Wang Chuanbao¹()

¹School of Mathematics and Statistics, Qingdao University, Shandong Qingdao 266071
²Department of Basic Courses, Guangzhou Maritime University, Guangzhou 510725

Received:2022-08-15 Revised:2023-04-10 Online:2023-10-26 Published:2023-08-09
Contact: Fei Chen E-mail:feichenstudy@163.com;shuai172021@163.com;yongyezhao@163.com;wcb1216@163.com
Supported by:
NSFC(12101345);Natural Science Foundation of Shandong Province of China(ZR2021QA017);Basic and Applied Basic Research Project of Guangzhou Basic Research Plan(202102020283)

摘要/Abstract

摘要：

该文主要研究 $\mathbb{R}^3$ 中可压缩磁流体力学方程大解的时间衰减率. 当 $(\sigma_{0}-1,u_{0},M_{0})$ 属于 $L^1\cap H^2$ 时, 基于 Chen[1] 等人的成果, 文献 [2] 中得到了 $\|\nabla(\sigma-1,u,M)\|_{H^1}\leqslant C(1+t)^{-\frac{5}{4}}$, 可见, 解二阶导数的时间衰减率不是最理想的. 因此, 该文通过借助频率分解[3] 的方法将 $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ 的时间衰减率改进为 $(1+t)^{-\frac{7}{4}}$.

关键词: 时间衰减率, 大解, 可压缩磁流体力学方程

Abstract:

This paper focus on time decay rate for large-solution about compressible magnetohydrodynamic equations in $\mathbb{R}^3$. Provided that $(\sigma_{0}-1,u_{0},M_{0})\in L^1\cap H^2$, based on the work of Chen et al.[1], $\|\nabla(\sigma-1,u,M)\|_{H^1}\leqslant C(1+t)^{-\frac{5}{4}}$ is obtained in reference [2], obviously, time decay rate of the 2nd-order derivative of the solution in [2] is not ideal. Here, we improve that of $\|\nabla^2 (\sigma-1,u,M)\|_{L^2}$ to be $(1+t)^{-\frac{7}{4}}$ by the frequency decomposition method[3].

Key words: Time decay rate, Large-solution, Compressible magnetohydrodynamic equations

中图分类号:

O175

陈菲,王帅,赵永叶,王传宝. 三维可压缩 MHD 方程大解的时间衰减率[J]. 数学物理学报, 2023, 43(5): 1397-1408.

Chen Fei,Wang Shuai,Zhao Yongye,Wang Chuanbao. Time Decay Rate for Large-Solution About 3D Compressible MHD Equations[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1397-1408.

参考文献

[1]	Chen Y H, Huang J C, Xu H Y. Global stability of large solutions of the 3-D compressible magnetohydrodynamic equations. Nonlinear Anal: Real World Appl, 2019, 47: 272-290
[2]	Wang S, Chen F, Wang C B. Estimation of decay rates to large-solutions of 3D compressible magnetohydrodynamic system. J Math Phys, 2022, 63: 111507
[3]	Wang W J, Wen H Y. Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion. Sci China Math, 2022, 65: 1199-1228
[4]	Fan J S, Yu W H. Strong solution to the compressible magnetohydrodynamic equations with vacuum. Nonlinear Anal: Real World Appl, 2009, 10(1): 392-409
[5]	Vol'pert A I, Hudjaev S I. On the Cauchy problem for composite systems of nonlinear equations. Math USSR-Sb, 1972, 16(4): 517
[6]	Hong G Y, Hou X F, Peng H Y, Zhu C J. Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum. SIAM J Math Anal, 2017, 49(4): 2409-2441
[7]	Wang D H. Large solutions to the initial-boundary value problem for planar magnetohydrodynamics. SIAM J Appl Math, 2003, 63(4): 1424-1441
[8]	Li H L, Xu X Y, Zhang J W. Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum. SIAM J Math Anal, 2013, 45(3): 1356-1387
[9]	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: 595-629
[10]	Hu X P, Wang D H. Global solutions to the three-dimensional full compressible magnetohydrodynamics flows. Commun Math Phys, 2008, 283: 255-284
[11]	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
[12]	Liu S Q, Yu H B, Zhang J W. Global weak solutions of 3D compressible MHD with discontinuous initial data and vacuum. J Differ Equ, 2013, 254(1): 229-255
[13]	Suen A. Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations. J Differ Equ, 2019, 268(6): 2622-2671
[14]	Suen A, Hoff D. Global low-energy weak solutions of the equations of three-dimensional compressible magnetohydrodynamics. Arch Rational Mech Anal, 2012, 205: 27-58
[15]	Wu G C, Zhang Y H, Zou W Y. Optimal time-decay rates for the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. J London Math Soc, 2021, 103(3): 817-845
[16]	Bie Q Y, Wang Q R, Yao Z A. Global well-posedness of the 3D incompressible MHD equations with variable density. Nonlinear Anal: Real World Appl, 2019, 47: 85-105
[17]	Chen F, Guo B L, Zhai X P. Global solution to the 3-D inhomogeneous incompressible MHD system with discontinuous density. Kinet Relat Mod, 2019, 12(1): 37-58
[18]	Chen F, Li Y S, Xu H. Global solution to the 3D nonhomogeneous incompressible MHD equations with some large initial data. Discret Contin Dyn Syst, 2016, 36(6): 2945-2967
[19]	Zhai X P, Li Y S, Yan W. Global well-posedness for the 3-D incompressible inhomogeneous MHD system in the critical Besov spaces. J Math Anal Appl, 2015, 432(1): 179-195
[20]	Zhai X P, Li Y S, Yan W. Well-posedness for the three dimension magnetohydrodynamic system in the anisotropic Besov spaces. Acta Appl Math, 2016, 143(1): 1-13
[21]	Chen Q, Tan Z. Global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations. Nonlinear Anal: Theory Methods Appl, 2010, 72(12): 4438-4451
[22]	Li F C, Yu H J. Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations. Proc Roy Soc Edinb Sect A Math, 2, 2011, 141(1): 109-126
[23]	Zhang J W, Zhao J N. Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics. Commun Math Sci, 2010, 8(4): 835-850
[24]	Gao J C, Chen Y H, Yao Z A. Long-time behavior of solution to the compressible magnetohydrodynamic equations. Nonlinear Anal: Theory Methods Appl, 2015, 128: 122-135
[25]	Pu X K, Guo B L. Global existence and convergence rates of smooth solutions for the full compressible MHD equations. Z Angew Math Phys, 2013, 64: 519-538
[26]	Gao J C, Tao Q, Yao Z A. Optimal decay rates of classical solutions for the full compressible MHD equations. Z Angew Math Phys, 2016, 67(2): 1-22
[27]	Tan Z, Wang H Q. Optimal decay rates of the compressible magnetohydrodynamic equations. Nonlinear Anal: Real World Appl, 2013, 14(1): 188-201
[28]	Huang W T, Lin X Y, Wang W W. Decay-in-time of the highest-order derivatives of solutions for the compressible isentropic MHD equations. J Math Anal Appl, 2021, 502(2): 125273
[29]	Zhu L M, Zi R Z. Decay estimates of the smooth solution to the compressible magnetohydrodynamic equations on $\mathbb{T}^3$. J Differ Equ, 2021, 288: 1-39
[30]	Shi W X, Zhang J Z. A remark on the time-decay estimates for the compressible magnetohydrodynamic system. Appl Anal, 2021, 100(11): 2478-2498
[31]	Wei R Y, Li Y, Guo B L. Global existence and convergence rates of solutions for the 3D compressible magnetohydrodynamic equations without heat conductivity. Appl Anal, 2020, 99(10): 1661-1684
[32]	Gao J C, Wei Z Z, Yao Z A. Decay rate of strong solution for the compressible magnetohydrodynamic equations with large initial data. Appl Math Lett, 2020, 102: 106100
[33]	王帅, 陈菲, 王传宝. 三维可压缩磁流体力学方程的重要估计. 理论数学, 2022, 12(8): 1305-1311
[33]	Wang S, Chen F, Wang C B. The important estimates for compressible MHD equations. Pure Mathematics, 2022, 12(8): 1305-1311