Solyanik estimates in harmonic analysis

Let $\mathcal{B}$ denote a collection of open bounded sets in $\mathbb{R}^n$, and define the associated maximal operator $M_{\mathcal{B}}$ by $$M_{\mathcal{B}}f(x) := \sup_{x \in R \in \mathcal{B}} \frac{1}{|R|}\int_R |f|.$$ The sharp Tauberian constant of $M_{\mathcal{B}}$ associated to $\alpha$, denoted by $C_{\mathcal{B}}(\alpha)$, is defined as $$C_{\mathcal{B}}(\alpha) := \sup_{E :\, 0 < |E| < \infty}\frac{1}{|E|}\big|\big\{x \in \mathbb{R}^n:\, M_{\mathcal{B}}\chi_E (x) > \alpha\big\}\big|.$$ Motivated by previous work of A. A. Solyanik, we show that if $M_{\mathcal{B}}$ is the uncentered Hardy-Littlewood maximal operator associated to balls, the estimate $$\lim_{\alpha \rightarrow 1^-}C_{\mathcal{B}}(\alpha) = 1$$ holds. Similar results for iterated maximal functions are obtained, and open problems in the field of Solyanik estimates are also discussed.Comment: 17 pages, 2 figures, minor typos corrected, to appear in Springer Proceedings in Mathematics & Statistic

Absence of squirt singularities for the multi-phase Muskat problem

In this paper we study the evolution of multiple fluids with different constant densities in porous media. This physical scenario is known as the Muskat and the (multi-phase) Hele-Shaw problems. In this context we prove that the fluids do not develop squirt singularities.Comment: 16 page

On the existence of stationary splash singularities for the Euler equations

In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.Comment: 19 page

Mixing solutions for the Muskat problem

We prove the existence of mixing solutions of the incompressible porous media equation for all Muskat type $H^5$ initial data in the fully unstable regime.Comment: 88 pages, 2 figure