Endpoint bilinear estimates and applications to the two-dimensional Poisson–Nernst–Planck system

We study the Cauchy problem of the two-dimensional Poisson–Nernst–Planck (PNP) system in Besov spaces $\dot{B}^{-3/2,r}_{4}$ for r ⩾ 2. Our work shows a dichotomy of well-posedness and ill-posedness depending only on r. Specifically, when r = 2, combining the key bilinear estimates in $L^{2}_T\dot{\mathcal{W}}^{-{1}/{2},4}\cap L^4_T\dot{\mathcal{W}}^{-1,4}$ with the heat semigroup characterization of Besov spaces, we prove the well-posedness of the PNP in $\dot{B}^{-3/2,2}_{4}$, while for r > 2 we show that the PNP is ill-posed in $\dot{B}^{-3/2,r}_{4}$ in the sense that the difference of the charges must satisfy certain requirements, i.e. either the difference belongs to $\dot{B}^{-3/2,r}_{4}$ for r > 2 and the summation belongs to $\dot{B}^{-3/2,2}_4$, or the difference belongs to $\dot{B}^{-3/2,2}_{4}$ and the summation belongs to $\dot{B}^{-3/2,r}_4$ for r > 2. Thus our results indicate that the difference of charges plays a crucial role and might provide some instability criterion for numerical analysis.