Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation
Abstract
We prove that the solution to the Córdoba-Córdoba-Fontelos equation θt−uθx=0, u=Hθ, on the real line with analytic initial datum θ0(x)=(1+x2)−1 develops a finite-time singularity while remaining uniformly bounded in C2/5. We also prove that, as the solution approaches the singular time, the C2/3 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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BibTeX
@misc{castro2026hölder,
title = {{Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation}},
author = {Ángel Castro and Alberto Enciso and Antonio J. Fernández},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37232},
eprint = {2609.37232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37232},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Hölder estimates up to the singular time for the Córdoba-Córdoba-Fontelos equation
AU - Ángel Castro
AU - Alberto Enciso
AU - Antonio J. Fernández
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37232
UR - https://arxiv.org/abs/2609.37232
AB - 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.
ER -