Global existence of a non-local semilinear parabolic equation with advection and applications to shear flow

TL;DR

This paper investigates the global existence of solutions to a non-local semilinear parabolic equation with advection on a torus, demonstrating solutions' existence under specific flow conditions such as mixing and shear flows.

Contribution

It establishes the local existence of mild solutions for arbitrary initial data and proves global solutions' existence for particular flow types, advancing understanding of non-local PDEs with advection.

Findings

Local existence of mild solutions for arbitrary data in L^2.

Global solutions exist under mixing flow conditions.

Global solutions exist under shear flow conditions.

Abstract

In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-\Delta u=|u|^p-\int_{\mathbb T^N} |u|^p \quad & \textrm{on} \quad \mathbb T^N, \\ \\ u \ \textrm{periodic} \quad & \textrm{on} \quad \partial \mathbb T^N \end{cases} \end{equation*} with initial data $u_0$ defined on $\mathbb T^N$. Here $v$ is an incompressible flow, and $\mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.