Edmunds-Evans essential spectra of the generalized Constantin-Lax-Majda linearization
Abstract
For 0<a<1, we study the linearization La about a smooth odd self-similar collapse profile (Ω,cl) of the generalized Constantin-Lax-Majda equation, where cl is the focusing exponent. On the origin-H2 realization X={φ∈ L2(0,∞):φ(0)=0,φ''∈ L2(0,∞)}, 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: Reλ=F=−1+cl/2 at infinity and Reλ=O=−c̃/2 at the origin, where c̃=cl+a(HΩ)(0) and H is the Hilbert transform. Off these lines, La−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/(848−9π2), 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.
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BibTeX
@misc{xu2026edmundsevans,
title = {{Edmunds-Evans essential spectra of the generalized Constantin-Lax-Majda linearization}},
author = {Jie Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13220},
eprint = {2609.13220},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13220},
}
RIS
TY - UNPB
TI - Edmunds-Evans essential spectra of the generalized Constantin-Lax-Majda linearization
AU - Jie Xu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.13220
UR - https://arxiv.org/abs/2609.13220
AB - 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∈ L^2(0,∞):φ(0)=0,φ''∈ L^2(0,∞)}, 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: Reλ=F=−1+c_l/2 at infinity and Reλ=O=−c̃/2 at the origin, where c̃=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.
ER -