论文标题
在三个基本域的IPM方程中准线性分层密度的定量渐近稳定性
Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the three fundamental domains
论文作者
论文摘要
我们分析了$ \ bbr^2 $在$ \ bbr^2 $相对于浮力频率$ n $上的2D无压缩多孔培养基方程中准线性分层密度的渐近稳定性。我们的分层目标密度是具有其斜率$ n $的大型背景线性轮廓的总和，以及可能是非线性和非单调的小扰动。 $ n $中的量化将不仅在允许的初始密度干扰的大小上进行，而且还将对目标密度偏离纯线性密度分层而不会失去其稳定性的程度。 For the purely linear density stratification, our method robustly applies to the three fundamental domains $\bbR^2,$ $\bbT^2,$ and $\bbT\times[-1,1]$, improving both the previous result by Elgindi (On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation, Archive for Rational Mechanics and Analysis, 225（2），573-599，2017）在$ \ bbr^2 $和$ \ bbt^2 $上，以及Castro-Córdoba-lear的研究（全球存在于受限IPM方程的准分类解决方案。 $ \ bbt \ times [-1,1] $。 $ \ bbr^2 $上分层密度的时间衰减率，以及新发现的渐近密度概要文件，$ \ bbt^2 $和$ \ bbt \ times [-1,1] $都很清晰，完全实现了线性化系统的水平。对于任何整数$ m \ geq 4 $，我们要求初始干扰在$ h^m $中很小，我们甚至通过合适的各向异性换向器估算值为任何正数$ m> 3 $放松。
We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on $\bbR^2$ with respect to the buoyancy frequency $N$. Our target density of stratification is the sum of the large background linear profile with its slope $N$ and the small perturbation that could be both non-linear and non-monotone. Quantification in $N$ will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability. For the purely linear density stratification, our method robustly applies to the three fundamental domains $\bbR^2,$ $\bbT^2,$ and $\bbT\times[-1,1]$, improving both the previous result by Elgindi (On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation, Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on $\bbR^2$ and $\bbT^2$, and the study by Castro-Córdoba-Lear (Global existence of quasi-stratified solutions for the confined IPM equation. Archive for Rational Mechanics and Analysis, 232(1), 437-471, 2019) on $\bbT\times[-1,1]$. The obtained temporal decay rates to the stratified density on $\bbR^2$ and to the newly found asymptotic density profiles on $\bbT^2$ and $\bbT\times[-1,1]$ are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in $H^m$ for any integer $m\geq 4$, which we even relax to any positive number $m>3$ via a suitable anisotropic commutator estimate.
