2026/07/22 by Jie Xu
#physics.flu-dyn #math-ph #math.MP
We give a spectral description of the self-similar collapse profile of the Constantin-Lax-Majda (CLM) equation, the a=0 anchor of the generalized family wt + a u wx = ux w, ux = Hw. Linearizing about the exact profile Ω(y) = -y/(y2+1/4) and realizing L0 as a closed operator on the origin-H2 space, we prove three things at a=0. Its essential spectrum meets the closed half-plane \Re λ≥ -1/2\ in the single vertical line \Re λ= -1/2\: the line is placed by a log-widening Weyl sequence, and an explicit Hardy-Mellin resolvent bound constructively empties the rest of the half-plane apart from 0 and 1. Its full point spectrum over ℂ, on the odd realization, is exactly \0,1\, the scaling and time-shift symmetry modes, with no embedded eigenvalues; removing these by the standard modulation leaves a spectral gap of 1/2 on X. The linear semigroup and its exact decay rate e-τ/2 are computed in closed form, but on a weighted space of the conjugated variable reached from X by a bounded transfer map; we keep the two separate, since L0 is non-normal and a spectral gap does not by itself give a decay rate in the X norm. A realization dichotomy identifies the in-strip smear of generic discretizations as the faithful spectrum of the maximal L2 realization, which origin-H2 removes. For a>0 we prove a conditional two-line inclusion for each admissible smooth focusing profile, recompute the branch cl(a) of Lushnikov, Silantyev, and Siegel as a cross-check, and record the formal scaling-relevance exponent s^*(a) = 1/cl(a), below which fractional dissipation is asymptotically subdominant in self-similar variables for fixed sufficiently regular data. The contribution is the realization-dependent spectral picture of the collapse profile itself.