本项目包含两个子课题:一个课题是致力于几何分析中的一些共形几何流的研究。基于我们已有的工作（Invent. Math., 187 (2012), no.2, 395-506； J. Func. Anal. 261 (2011), 934-980）和其他人的工作,我们集中于低维标准球面情形的纯量曲率流的研究以及所有偶数维标准球面上的Q-curvature流的研究。此外,我们期待能够在著名的"四阶Yamabe问题"上做出一些有益的尝试。另外一个课题是致力于向列液晶问题中所诱导的非线性波方程(组)的研究。对于此方程(组)的Cauchy问题,数学上最令人感兴趣的问题是证明一维空间变量情形下的全局弱解的存在性以及高维(二维、三维)空间变量情形下的短时间解的存在性和正则性。

This project includes two topics: one is concerned with some conformal geometric flows in geometric analysis. Building on our previous works (Invent. Math., 187 (2012), no.2, 395-506； J. Func. Anal. 261（2011,934-980) and others',we focus on the scalar curvature flow on the round spheres of low dimensions and Q-curvature flow on all even dimensional round spheres by admitting changing signs curvature candidates. Moreover, we expect to do some efficient attempts to the famous "fourth order Yamabe problem" in conformal geometry. The other one is devoted to the study of nonlinear wave equation/system, which arises from the investigations on nematic liquid crystals. For the Cauchy problem of this equation/system, the most interesting problem in mathematics is to prove the existence of global weak solutions in the case of 1-D space variables and the existence and regularities of short time solutions in the case of 2-D or 3-D space ones.