×

Endpoint bilinear estimates and applications to the two-dimensional Poisson-Nernst-Planck system. (English) Zbl 1278.35036

Summary: We study the Cauchy problem of the two-dimensional Poisson-Nernst-Planck (PNP) system in Besov spaces \(\dot{B}^{-3/2,r}_{4}\) for \(r \geq 2\). 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.

MSC:

35B45 A priori estimates in context of PDEs
35K45 Initial value problems for second-order parabolic systems
42B37 Harmonic analysis and PDEs
35R10 Partial functional-differential equations
35K58 Semilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI