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Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations
Journal of the London Mathematical Society ( IF 1.2 ) Pub Date : 2020-10-20 , DOI: 10.1112/jlms.12393
Guochun Wu 1 , Yinghui Zhang 2 , Weiyuan Zou 3
Affiliation  

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 L 2 . 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 L r ‐norm with 2 r and the first‐order derivatives of the velocity and magnetic field in L 2 ‐norm when the L 1 ‐norm of the initial perturbation is bounded. Moreover, we also show the lower bounds on the rates of decay.

中文翻译:

具有不连续的初始数据和大振荡的3D可压缩磁流体动力流的最佳时间衰减率

本文关注具有不连续初始数据和大振动的3D可压缩磁流体动力流的弱解的时衰减率。到3D可压缩磁流体流动的Cauchy问题弱解的整体存在已经由孙-霍夫(建立拱。理性。机甲元素分析。205(2012)27-58)和孙(J.微分方程268(2020 )2622-2671)(另请参见Liu,Yu和Zhang J.微分方程254(2013)229-255),在初始能量适当较小的情况下,初始密度为正且基本上有界,且初始梯度为速度和磁场限制在 大号 2个 。但是,据我们所知,到目前为止,此类解决方案的时间衰减率尚无结果。本文的主要新颖之处在于对这个问题给出了积极的回应。更准确地说,我们获得了解决方案的最佳时间衰减率 大号 [R -规范 2个 [R 和速度和磁场的一阶导数 大号 2个 -规范何时 大号 1个 -初始扰动的范数是有界的。此外,我们还显示了衰减率的下限。
更新日期:2020-10-20
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