The spectral picture of self-similar collapse in the Constantin-Lax-Majda equation

Abstract

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 $w_t + a\,u\,w_x = u_x\,w$, $u_x = Hw$. Linearizing about the exact profile $\Omega(y) = -y/(y^2+1/4)$ and realizing $L_0$ as a closed operator on the origin-$H^2$ space, we prove three things at $a=0$. Its essential spectrum meets the closed half-plane $\{\mathrm{Re}\,\lambda \ge -1/2\}$ in the single vertical line $\{\mathrm{Re}\,\lambda = -1/2\}$: 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 $\mathbb{C}$, 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 $e^{-\tau/2}$ 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 $L_0$ 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 $L^2$ realization, which origin-$H^2$ removes. For $a>0$ we prove a conditional two-line inclusion for each admissible smooth focusing profile, recompute the branch $c_l(a)$ of Lushnikov, Silantyev, and Siegel as a cross-check, and record the formal scaling-relevance exponent $s^*(a) = 1/c_l(a)$, 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.