On the regularity of the solution map of the porous media equation

TL;DR

This paper investigates the regularity of the solution map for the incompressible porous media equation in Sobolev spaces, showing it is nowhere locally uniformly continuous while particle trajectories are analytic.

Contribution

It demonstrates the irregularity of the solution map in Sobolev spaces and establishes the analyticity of particle trajectories for the porous media equation.

Findings

Solution map is nowhere locally uniformly continuous in Sobolev spaces.

Particle trajectories are analytic curves in .

Provides insights into the regularity properties of solutions.

Abstract

In this paper we consider the incompressible porous media equation in the Sobolev spaces $H^s(\R^2), s > 2$. We prove that for $T > 0$ the time $T$ solution map $\rho_0 \mapsto \rho(T)$ is nowhere locally uniformly continuous. On the other hand we show that the particle trajectories are analytic curves in $\R^2$.