On the generalized porous medium equation in Fourier-Besov spaces

TL;DR

This paper investigates a generalized porous medium equation with fractional Laplacian in Fourier-Besov spaces, establishing local and global well-posedness results, and providing a blowup criterion for solutions.

Contribution

It introduces new well-posedness results for a broad class of fractional porous medium equations in Fourier-Besov spaces, including conditions for global existence and blowup.

Findings

Local well-posedness in Fourier-Besov spaces for large initial data

Global existence for small initial data

Blowup criterion for solutions

Abstract

We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term. For a large class of equations corresponding to the form: $u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu)$, we get their local well-posedness in Fourier-Besov spaces for large initial data. If the initial data is small, then the solution becomes global. Furthermore, we prove a blowup criterion for the solutions.