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

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 (Formula presented) with 1 ≤ p, σ ≤ ∞ 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.