三维可压缩Navier-Stokes-Fourier方程组的精确时间衰减率  

The sharp time decay rate of the three-dimensional compressible Navier-Stokes-Fourier system

作　　者：陈玉惠 潘荣华 Yuhui Chen;Ronghua Pan

机构地区：[1]中山大学航空航天学院,深圳518107 [2]School of Mathematics,Georgia Institute of Technology,Atlanta,GA 30332,USA

出　　处：《中国科学:数学》2025年第3期605-634,共30页Scientia Sinica:Mathematica

基　　金：国家重点研发计划(批准号:2023YFA1010300);美国国家自然科学基金(批准号:DMS-1813603和DMS-2108453)资助项目。

摘　　要：本文研究三维可压缩Navier-Stokes-Fourier方程组在全空间中的解U(t,x)趋近恒稳态U=(ρ>0,0,θ>0)的精确时间衰减率.当初始值是在U附近的光滑小扰动时,本文通过证明U(t,x)与根据初始扰动演化的相应线性化问题的解Ue(t,x)具有相同的衰减率来实现这一目标.在初始扰动满足一些温和的非退化条件下,本文证明了U−Ue的L^(2)-模至少以(1+t)^(−5/4)的速率衰减,比Ue的(1+t)^(−3/4)的速率衰减得更快.本文的方法结合了从谱分析中得到的线性最优时间衰减率(上界和下界)和巧妙的能量方法.In this paper,we are concerned with the sharp time decay rate of the solution U(t,x)for the compressible Navier-Stokes-Fourier system for x∈R 3 to the constant steady state U=(ρ>0,0,θ>0)when the initial data are small smooth perturbations near U.This is achieved by proving that U(t,x)shares the same decay rate with Ue(t,x),the solution to the corresponding linearized problem for the evolution of the initial perturbation.Under a mild non-degenerate condition on the initial perturbations,we show that the L^(2)-norm of U−Ue decays at least at the rate of(1+t)^(−5/4),which is faster than the rate(1+t)^(−3/4) for Ue.Our method combines the linear sharp time decay rate(for both upper and lower bounds)obtained from the spectral analysis and the carefully designed energy method.

关 键 词：可压缩Navier-Stokes-Fourier方程组 精确衰减率 谱分析 能量方法 

分 类 号：O175[理学—数学]