Edmunds-Evans essential spectra of the generalized Constantin-Lax-Majda linearization

Abstract

For $0<a<1$, we study the linearization $L_a$ about a smooth odd self-similar collapse profile $(\Omega ,c_l)$ of the generalized Constantin-Lax-Majda equation, where $c_l$ is the focusing exponent. On the origin-$H^2$ realization $X=\{\phi \in L^2(0,\infty ):\phi (0)=0,{\phi }^{\prime \prime }\in L^2(0,\infty )\}$, 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: $\mathrm{Re}\lambda =F=-1+c_l/2$ at infinity and $\mathrm{Re}\lambda =O=-\tilde{c}/2$ at the origin, where $\tilde{c}=c_l+a(H\Omega )(0)$ and $H$ is the Hilbert transform. Off these lines, $L_a-\lambda$ 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 $0<a<400/(848-9{\pi }^2)$, 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.

arXiv comment50 pages, 1 figure