Ill-posedness issue for the drift diffusion system in the homogeneous Besov spaces

Abstract

We consider the ill-posedness issue for the drift-diffusion system of bipolar type by showing that the continuous dependence on initial data does not hold generally in the scaling invariant Besov spaces. The scaling invariant Besov spaces are $\dot{B}_{p, \sigma}^{-2+ n/p} (\mathbb{R}^{n})$ with $1 \leq p, \sigma \leq \infty$ and we show the optimality of the case $p = 2n$ to obtain the well-posedness and the ill-posedness for the drift-diffusion system of bipolar type.