The sharp time-decay rates for one-dimensional compressible isentropic Navier-Stokes and magnetohydrodynamic flows  

作　　者：Yuhui Chen Minling Li Qinghe Yao Zheng-an Yao 

机构地区：[1]School of Aeronautics and Astronautics,Sun Yat-sen University,Guangzhou 510275,China [2]School of Mathematics,Sun Yat-sen University,Guangzhou 510275,China

出　　处：《Science China Mathematics》2023年第3期475-502,共28页中国科学：数学（英文版）

基　　金：supported by Guangdong Basic and Applied Basic Research Foundation(Grant No.2019A1515110733);National Natural Science Foundation of China(Grant Nos.11971496 and 11972384);National Key R&D Program of International Collaboration(Grant No.2018YFE9103900);National Key R&D Program of China(Grant No.2020YFA0712500)。

摘　　要：In this paper, we investigate the large-time behavior of strong solutions to the Cauchy problem for one-dimensional compressible isentropic magnetohydrodynamic equations near a stable equilibrium. The difference between the one-dimensional and multi-dimensional cases is a feature for compressible flows and also brings new difficulties. In contrast to the multi-dimensional case, the decay rates of nonlinear terms may not be faster than linear terms in dimension one. To handle this, we shall present a new energy estimate in terms of a combination of the solutions with small initial data. We aim to establish the sharp upper and lower bounds on the L^(2)-decay rates of the solutions and all their spatial derivatives when the initial perturbation is small in L^(1)(R) ∩ H^(2)(R). It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solutions so that the large-time behavior for the hyperbolic-parabolic system is exactly sharp. As a byproduct, the above result is also valid for compressible Navier-Stokes equations. Our approach is based on various interpolation inequalities, energy estimates, spectral analysis, and Fourier time-splitting and high-low frequency decomposition methods.

关 键 词：compressible Navier-Stokes equations magnetohydrodynamic equations optimal decay rates upper bound lower bound 

分 类 号：O175[理学—数学]