两类迁移扩散方程组的若干问题研究

项目名称： 两类迁移扩散方程组的若干问题研究

项目编号： No.11301228

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 邓超

作者单位： 江苏师范大学

项目金额： 22万元

中文摘要： 最近数值研究表明Poisson-Nernst-Planck(PNP)模型对一些初值稳定而对另一些初值不稳定。此外,Bourgain-Pavlovic在[J.Funct.Anal.,255(2008),2233-2247]中构造了一类能量无限的周期初值用以证明Navier-Stokes方程组的不适定性。本项目拟使用周期初值函数的光滑截断证明PNP模型在全空间中的不适定性或不稳定性。内容包括：建立PNP模型的端点双线性估计；构造适当范数有界的非周期初值并结合双线性估计证明PNP模型的不稳定性;分析使PNP模型不稳定的初值和本项目拟构造的初值之间的差异性质;直观上Keller-Segel(KS)模型是PNP模型的特例,应具有更好的性质。因此项目将研究KS在BMO-2空间中的适定性和更大的函数空间中的不稳定性;比较PNP和KS的不稳定性理论,加深人们对PNP和KS的非线性结构的内在差异的认识。

中文关键词： Poisson-Nernst-Planck模型；Keller-Segel模型；分数阶扩散方程；适定性；不适定性

英文摘要： Recent numerical results indicate that for some data the Poisson-Nernst-Planck system (PNP) is stable while for some other data it is unstable. By checking Bourgain-Pavlovic's ill-posedness results [cf. J.Funct.Anal., 255(2008),2233-2247], we observe that the datum constructed by Bourgain etc. are periodic functions with infinite energy in the whole space domain. To get some norm finite data, one way is to cut off the periodic datum. In this project, we plan to study the instability of PNP system by constructing some norm finite initial data. Precisely, at first, we establish several key endpoint bilinear estimates; then by combining the bilinear estimates with the constructed data, we prove instability of the PNP system. Additionally, we check whether the data appears in numerical simulations satisfies the same property of the constructed data. It is clear that Keller-Segel (KS) system can be thought as a special case of the PNP system. Hence KS system should have better property than the PNP system. As a consequence, we aim at establishing well-posedness for the KS system in BMO-2 space and instability for the KS system in function spaces which are larger than BMO-2. Finally, by checking the difference of their instability results, we have better understanding of their different structures.

英文关键词： Poisson-Nernst-Planck system；Keller-Segel system；fractional diffusion system；well-posedness；ill-posedness