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

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 abla 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 $L^2$.