Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations
Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations
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DOI：
10.1112/jlms.12393
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
Guochun Wu;Yinghui Zhang;Weiyuan Zou
中科院分区：
文献类型：
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作者：
Guochun Wu;Yinghui Zhang;Weiyuan Zou
本文研究了具有间断初值和大振荡的三维可压缩磁流体动力学方程弱解的时间衰减率。Suen-Hoff（Arch. Ration.机械肛门205（2012）27-58）和孙（J.微分方程268（2020）2622-2671）（也参见Liu，Yu和Zhang J. Differential Equations 254（2013）229-255），在初始能量适当小的条件下，初始密度为正且基本有界，初始速度梯度和磁场梯度在L2内有界。然而，据我们所知，到目前为止，还没有关于这种解决方案的时间衰减率的结果。本文的主要新奇在于对这一问题给予了积极的回应。更精确地说，当初始扰动的L1-范数有界时，我们得到了Lr-范数下2 <$r <$∞解的最优时间衰减率以及L2-范数下速度和磁场的一阶导数。此外，我们还显示了衰减率的下限。
This paper is concerned with time‐decay rates of the weak solutions to the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. The global existence of weak solutions to the Cauchy problem of the 3D compressible magnetohydrodynamic flows has been established by Suen–Hoff (Arch. Ration. Mech. Anal. 205 (2012) 27–58) and Suen (J. Differential Equations 268 (2020) 2622–2671) (also see Liu, Yu and Zhang J. Differential Equations 254 (2013) 229–255), under the condition that the initial energy is suitably small, the initial density is positive and essentially bounded and the gradients of initial velocity and magnetic field are bounded in L2 . However, to our best knowledge, so far there is no result on time‐decay rates of such solutions. The main novelty of this paper is to give a positive response to this problem. More precisely, we obtain the optimal time‐decay rates of the solutions in Lr ‐norm with 2⩽r⩽∞ and the first‐order derivatives of the velocity and magnetic field in L2 ‐norm when the L1 ‐norm of the initial perturbation is bounded. Moreover, we also show the lower bounds on the rates of decay.