二维Kuramoto-Sivashinsky方程中的非线性不稳定性
Nonlinear Instability in the 2D Kuramoto-Sivashinsky equation
AI总结：
研究二维Kuramoto-Sivashinsky方程在特定环面上非线性不稳定性，通过考虑最大增长模式\(\lambda_0\)，利用近似解构造、不动点论证及延拓论证，证明存在小尺寸初始数据流形，使解在特定时间尺度上有特定变化。
AI中文摘要：
本文分析了Kuramoto-Sivashinsky方程（KSE），它是火焰前沿传播的模型，在任意大小为\(2L\)（\(L>\pi\)）的二维方形环面上。在这种情况下，原点处的线性化方程有有限数量的增长模式，对应于线性算子\(-\Delta^2-\Delta\)的正特征值。二维KSE在两个空间维度上解的长时间行为问题在很大程度上仍未解决。本文主要目的是在非线性层面分析围绕增长模式的不稳定性。考虑最大增长模式\(\lambda_0\)，证明存在任意小尺寸为\(\varepsilon\)的有限维初始数据流形，使得相应解在\({\rm log}(\varepsilon^{-1})\)量级的时间尺度上变为\(O(1)\)大小。证明基于从最大线性增长模式分叉的近似解的精确量化构造、带指数权重的不动点论证以构造局部时间解以及基于精确能量估计和准微分演算的延拓论证。
英文摘要：
In this paper we analyze the Kuramoto-Sivashinsky equation (KSE), a model of flame-front propagation, on a two-dimensional square torus of arbitrary size $2 L$ with $L > π$. In this case, the linearized equation at the origin admits a finite number of growing modes, which corresponds to the positive eigenvalues of the linear operator $- Δ^2 - Δ$. The problem of analyzing the long-time behavior of solutions of the 2D KSE in two spatial dimensions remains largely open; the only global existence results are for sufficiently small tori, or for sufficiently anisotropic and thin domains, due to the lack of good a priori estimates. The main purpose of this paper is to analyze the instability around growing modes at the nonlinear level. More precisely, we consider the maximal growing mode $λ_0$ and we show that there is a finite dimensional manifold of initial data of size $\varepsilon$ arbitrarily small such that the corresponding solutions become of size $O(1)$ over a time-scale of order ${\rm log}(\varepsilon^{- 1})$. The proof is based on several ingredients such as a sharp quantitative construction of an approximate solution bifurcating from the maximal linearly growing mode, a fixed point argument with exponential weights to construct local in time solutions, and a continuation argument based on sharp energy estimates and para-differential calculus.
