Gross-Piteavskii方程组解的相分离现象研究

The most commonly used model for the description of Bose-Einstein condensates, the Gross-Pitaevskii equations, is now supported both by data from extensive experiments and by a solid mathematical theory. BEC has its important physical background, therefore GP systems have received great concerns in recent years...For the singular limits, different components of the solutions to GP equations, which describe multi-component rotating BEC, are spatially segregated，and then produce a free boundary problem. We plan to study two problems in this project:..(1) Using some monotonicity formulas and Liouville theorems，we want to obtain the uniform Holder bounds of the solutions of the model, ..(2) We study the regularity of the free boundary of related limiting equations and also segregation patterns... Through the research of these problems about Gross-Pitaevskii equations, we want to know more about Bose-Einstein condensates.

经过大量的试验数据和严格的数学理论论证，Gross-Pitaevskii（GP）系统被平凡用以描述Bose-Einstein凝聚（BEC）现象。由于BEC模型重要的物理背景，GP系统的相关研究近年来受到人们的广泛关注。.　　当参数足够大时，描述多组份旋转BEC的GP方程组复值解的奇异极限在空间上支集分离，由此产生自由边界问题。本项目我们计划关于GP系统研究两个方面的问题：.（1）运用单调公式和Liouville型定理，我们希望证明系统复值解关于参数的一致Holder连续性，.（2）进一步利用几何测度理论讨论自由边界的正则性，以及空间分离模式。.　　对GP系统这些问题的研究，有利于我们更好地理解BEC现象。

本课题主要研究了Bose-Einstein凝聚及生物数学中描述多个物种竞争行为的数学模型中出现的相分离现象及其动力学行为, 另外还研究了若干带自由边界条件模型的动力学行为以及一个分数次Keller-Segel趋化模型的衰减估计，这些问题有强烈的物理和生物学背景，又与几何发展方程密切相关，是当前偏微分方程领域非常活跃的课题。研究这些问题能帮助我们更好地理解各个科目之间的关系。