Regularity and Capacity for the Fractional Dissipative Operator

TL;DR

This paper investigates the regularity and capacity properties of the fractional dissipative operator, providing sharp Hausdorff dimension estimates for blow-up sets in fractional dissipative equations.

Contribution

It establishes new analytic-geometric properties and sharp dimension estimates related to the fractional dissipative operator and its solutions.

Findings

Sharp Hausdorff dimension estimate for blow-up set

Analytic-geometric properties of fractional dissipative operator

Insights into regularity and capacity associated with the operator

Abstract

This note is devoted to exploring some analytic-geometric properties of the regularity and capacity associated to the so-called fractional dissipative operator $\partial_t+(-\Delta)^\alpha$, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation $(\partial_t+(-\Delta)^\alpha)u(t,x)=F(t,x)$ subject to $u(0,x)=0$.