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A convergence result for the master operator

Wenxiong Chen, Yahong Guo, Congming Li, Yugao Ouyang

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 $b \equiv 0$.

A convergence result for the master operator

Abstract

In this paper, we establish a convergence result for the fully fractional heat operator , also known as the master operator, stated as follows: for some nonnegative constant . 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 can indeed be positive. This highlights a key distinction between nonlocal and local operators: for a local heat operator, such as , it is well-known that .

Paper Structure

This paper contains 3 sections, 2 theorems, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Convergence
  3. Counterexamples

Key Result

Theorem 1.1

Let $n\ge 1$, $s\in (0,1)$ and $\varepsilon >0$ sufficiently small. Suppose that is a sequence of nonnegative functions, with $\{R_i\}$ be a sequence of positive numbers converging to $+\infty$. If $\{u_i\}$ converges in $C^{2s+\varepsilon,s+\frac{\varepsilon}{2}} _{x,t,loc}(\mathbb{R} ^n\times\mathbb{R})$ to a function $u\in \mathcal{L}^{2s,s} (\mathbb{R} ^n\times\mathbb{R})$, then there exists

Figures (3)

  • Figure 1: 2-dim. case, partition of outer domain when $(x,t)=(0,0)$.
  • Figure 2: 2-dim. case, partition of outer domain when $t=0$.
  • Figure 3: Partition of $\mathbb{R}^2$ into regions $I_+^j$ and $I_-^j$.

Theorems & Definitions (7)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • proof : Proof of Theorem\ref{['main thm']}
  • proof : Proof of Theorem \ref{['main thm2']}