Remarks on the ill-posedness results for the drift diffusion system (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

The ill-posedness issue is considered for the drift-diffusion system of bipolar type. It is shown 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+frac{n}{p}(mathbb{R}^{n}) with 1 leq p, $sigma$leq 1, and the reason on the optimality of the case p=2n is explained to obtain the well-posedness and the ill-posedness for the drift diffusion system of bipolar type with the attention to the divergence form structure of the nonlinear term. The scaling invariant spaces for the heat equation with the nonlinear term u^{2} are same as those for the drift diffusion system and the difference is also indicated on the ill-posedness results between the heat equation and the drift diffusion system.