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 ${\theta }_t-u{\theta }_x=0$, $u=H\theta$, on the real line with analytic initial datum ${\theta }_0(x)=(1+x^2{)}^{-1}$ 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 $\theta$ 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.