A convergence result for the master operator

TL;DR

This paper proves a convergence result for the fully fractional heat operator, showing that under certain conditions, the operator applied to a sequence converges to the operator applied to the limit function plus a constant.

Contribution

It establishes a fundamental convergence result for the master operator, addressing key questions in blow-up analysis and distinguishing nonlocal from local operators.

Findings

Convergence of the master operator under function sequence limits.

Existence of a positive constant in certain convergence cases.

Highlights difference between nonlocal and local heat operators.

Abstract

In this paper, we establish a convergence result for the fully fractional heat operator $\ma{s}$, also known as the master operator, stated as follows: \[\mbox{If\ }u_i\to u\ \mbox{in}\ C^{2,1}_{x,t,loc}(\R^n\times\R),\ \mbox{then}\ \ma{s} u_i\to \ma{s}u-b\ \mbox{a.e. in}\ \R^n\times\R,\] for some nonnegative constant $b$. This result addresses a fundamental question in the blow-up and rescaling analysis, which are essential for establishing a priori estimates for solutions of master equations. Additionally, we present examples demonstrating that in certain cases, the constant $b$ can indeed be positive. This highlights a key distinction between nonlocal and local operators: for a local heat operator, such as $\partial_t - \lap$, it is well-known that $b \equiv 0$.