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Global Existence of a Non-local Semilinear Parabolic Equation with Advection and Applications to Shear Flow by Feng, Yu; Hu, Bingyang; Xu, Xiaoqian; Zhang, Yeyu is a scholarly article available to read on EtoBox.

What is Global Existence of a Non-local Semilinear Parabolic Equation with Advection and Applications to Shear Flow about?

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 $L^2$.

Author

Feng, Yu; Hu, Bingyang; Xu, Xiaoqian; Zhang, Yeyu

Published

2021

Language

EN