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arXiv · 2609.37232

Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation

Ángel Castro·Alberto Enciso·Antonio J. Fernández
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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.

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Ángel Castro, Alberto Enciso, Antonio J. Fernández. 2026-09-29. Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation. https://arxiv.org/abs/2609.37232

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