Self-Similar Blow-Up in the Navier-Stokes Equations: Spectral Structure, Eigenvalue Stability, and the L³ Cage

We investigate the spectral structure underlying discrete families of self-similar blow-up solutions to the three-dimensional Navier-Stokes and Euler equations. Recent computational work by Buckmaster, Cao, and Gómez-Serrano (2023) discovered families of asymptotically self-similar blow-up profiles for the Euler equations, indexed by an instability parameter n, with blow-up rates following the empirical formula λ_n ≈ 1/(an+b)+c — a discrete spectrum whose origin remained unexplained. We propose a spectral explanation: the reciprocal transformation μ_n = 1/(λ_n - c) reveals linear growth μ_n ≈ an+b, matching the Weyl asymptotic signature of eigenvalues of an elliptic operator on a bounded domain. Via Rabinowitz global bifurcation theory, each linear eigenvalue generates a branch of nonlinear self-similar solutions, connecting the observed discrete spectrum to classical spectral theory. We further establish that the critical L³ norm acts as a "cage" constraining blow-up dynamics: the L³ norm of any self-similar profile must equal exactly the norm of the singular steady state (-1)-homogeneous solution, creating a codimension-one constraint that discretizes the solution space. For the Navier-Stokes equations, we prove that viscous damping strictly contracts the spectral gaps, ensuring λ_n(ν) < λ_n(0) for all n, which implies that any Navier-Stokes self-similar blow-up requires overcoming a strictly larger stability barrier than its Euler counterpart. These results provide a unified framework connecting computational observations to classical PDE spectral theory and suggest concrete numerical tests for validation.