发表机构
University of Illinois Chicago（伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究广义Constantin-Lax-Majda方程线性化的Edmunds-Evans本质谱，确定其由两条垂直线及闭带构成，并证明实际坍缩轮廓实现非空Browder带。
AI 中文摘要
对于$0<a<1$，我们研究广义Constantin-Lax-Majda方程关于光滑奇自相似坍缩轮廓$(\Omega,c_l)$的线性化$L_a$，其中$c_l$为聚焦指数。在原点-$H^2$实现$X=\{\varphi\in L^2(0,\infty):\varphi(0)=0,\varphi''\in L^2(0,\infty)\}$上，在显式输运、正则性、加权导数及远场假设下，前三个Edmunds-Evans本质谱恰为两条可能重合的垂直线的并集：在无穷远处$\operatorname{Re}\lambda=F=-1+c_l/2$，在原点处$\operatorname{Re}\lambda=O=-\tilde c/2$，其中$\tilde c=c_l+a(H\Omega)(0)$，$H$为Hilbert变换。在这些线之外，$L_a-\lambda$为Fredholm算子，在外部区域指标为零，在$F<O$时的开带内指标为$+1$，在$O<F$时为$-1$。第四和第五个Edmunds-Evans谱及Browder本质谱等于闭带。开带中指标为$+1$的每一点都是特征值。Huang-Qin-Wang-Wei正平流构造中满足$0<a<400/(848-9\pi^2)$的每个不动点，在归一化后满足全部假设包。对所有足够小的正$a$，每个这样的不动点都有$F<O$。因此实际坍缩轮廓实现了非空Browder带。附加的核非退化条件(ND)给出精确的个体核与余核维数。我们在显式垂直能量带之外证明该条件，并且在足够高的频率下，在开带中与远场线分离的每个闭垂直子带上证明之。开带中的失败是局部有限的，且仅在高频下可能向该线累积。
英文摘要
For $0<a<1$, we study the linearization $L_a$ about a smooth odd self-similar collapse profile $(Ω,c_l)$ of the generalized Constantin-Lax-Majda equation, where $c_l$ is the focusing exponent. On the origin-$H^2$ realization $X=\{φ\in L^2(0,\infty):φ(0)=0,φ''\in L^2(0,\infty)\}$, under explicit transport, regularity, weighted-derivative, and far-field hypotheses, the first three Edmunds-Evans essential spectra are exactly the union of two possibly coincident vertical lines: $\operatorname{Re}λ=F=-1+c_l/2$ at infinity and $\operatorname{Re}λ=O=-\tilde c/2$ at the origin, where $\tilde c=c_l+a(HΩ)(0)$ and $H$ is the Hilbert transform. Off these lines, $L_a-λ$ is Fredholm, with index zero on the exterior components, $+1$ in the open strip when $F<O$, and $-1$ when $O<F$. The fourth and fifth Edmunds-Evans spectra and the Browder essential spectrum equal the closed strip. Every point of the open strip with index $+1$ is an eigenvalue. Every fixed point of the Huang-Qin-Wang-Wei positive-advection construction with $0<a<400/(848-9π^2)$, after normalization, satisfies the full hypothesis package. For all sufficiently small positive $a$, every such fixed point has $F<O$. Thus actual collapse profiles realize a nonempty Browder band. An additional kernel-nondegeneracy condition (ND) yields exact individual kernel and cokernel dimensions. We prove it outside an explicit vertical energy slab and, at sufficiently high frequency, on every closed vertical substrip of the open band separated from the far-field line. Failures in the open band are locally finite and can accumulate at high frequency only toward that line.
Comments50 pages, 1 figure
