可压缩等熵Navier-Stokes方程的点态衰减极限

中文摘要 AI 辅助

我们研究一维可压缩等熵Navier-Stokes方程中常状态的小型局部扰动的长时间点态行为。在减去两个Burgers扩散波、所有高阶扩散波的收敛和以及跨族粘性修正之后，我们证明了精确物理余项的保锥点态估计，特别地，\\[ |R_i(x,t)|\leq C E_N\log(2+t)\Psi_i(x,t), \qquad \sup_{x\in\mathbb R}\Psi_i(x,t)\leq C(1+t)^{-1}. \\] 这里\\(E_N\\)衡量初始数据的大小，\\(\Psi_i\\)是锥分解权重；两者都在下面的主定理中精确定义。因此\\(\\|R_i(t)\\|_{L^\infty}\leq C E_N(1+t)^{-1}\log(2+t)\\)。关键的新思想是对余项的空间反导数应用族wise Cole--Hopf变换，这精确地消除了临界同族一阶反馈。我们进一步构造了适应于两个特征族的近似Green函数，并将其与高斯模态提取和Kawashima型能量论证相结合。这产生了在极限衰减速率下的保锥估计，仅损失对数因子。

英文摘要

We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, \[ |R_i(x,t)|\leq C E_N\log(2+t)Ψ_i(x,t), \qquad \sup_{x\in\mathbb R}Ψ_i(x,t)\leq C(1+t)^{-1}. \] Here \(E_N\) measures the size of the initial data and \(Ψ_i\) is the cone-resolved weight; both are defined precisely in the main theorem below. Thus $\|R_i(t)\|_{L^\infty}\leq C E_N(1+t)^{-1}\log(2+t)$. The key new idea is to apply a familywise Cole--Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.

发表机构

• The Chinese University of Hong Kong（香港中文大学）
• Guangxi University（广西大学）
• South China Normal University（华南师范大学）

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