Singularity formation in the CCF equation with fractional dissipation (α ∈ [1/2, 1])

Determine whether finite-time singularity (blow-up from smooth initial data) occurs for the Córdoba–Córdoba–Fontelos (CCF) one-dimensional nonlocal transport equation when augmented with a fractional dissipation term (−Δ)^{α/2} for fractional exponents α in the range 1/2 ≤ α ≤ 1, thereby clarifying the dissipative threshold for blow-up in this model.

Background

The Córdoba–Córdoba–Fontelos (CCF) equation is a simplified nonlocal transport model closely connected to mechanisms relevant to 3D Euler and Navier–Stokes dynamics. It serves as a tractable setting to explore singularity formation and stability properties in active scalar equations.

The central issue is whether the addition of fractional dissipation (−Δ)^{α/2} prevents or permits finite-time blow-up. The range α ∈ [1/2, 1] lies in a regime where dissipation is comparatively strong, and determining if singularities still form is a longstanding problem highlighted in prior works, including an explicit listing as "Open Problem 1" in the literature referenced by the authors.

The present paper reports improved numerical blow-up profiles and new unstable solutions for the CCF model (without dissipation), which in turn suggest higher thresholds for admissible α below which blow-up persists; however, the fundamental question of whether blow-up survives fractional dissipation across the entire interval 1/2 ≤ α ≤ 1 remains unresolved.