Year: 2026
Type: preprint
Abstract: We prove that the solution to the Córdoba-Córdoba-Fontelos equation $θ_t-uθ_x=0$, $u=Hθ$, on the real line with analytic initial datum $θ_0(x)=(1+x^2)^{-1}$ develops a finite-time singularity while remaining uniformly bounded in $C^{2/5}$. We also prove that, as the solution approaches the singular time, the $C^{2/3}$ seminorms of both $θ$ and $u$ diverge. The main ingredient is an integral representation in terms of a probability measure that is preserved by the nonlinear evolution. From this representation we derive a pointwise quadratic estimate for the key nonlinear term in the equation for $u$, using a comparison of positive kernels. Together with evolution identities for the moments of the measure, this estimate yields both the uniform Hölder bound and the divergence of the higher seminorms.
Source: arXiv (Cornell University)
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G121809054, G1401168639, G143639328, G1701110065, G1855990336, G1908868068, G1972043705, G2309174360, G2425056411, G252646220, G2585926585, G2660406951, G279363373, G2967504892, G3518538256, G4233400527, G430777504, G5195188938, G5421974903, G5805502524, G5845768719, G6282578303, G7391727506, G745419941, G846842501, G8575141037, G897484643, G99296377
OpenAlex ID: W7214952420