非线性薛定谔方程和波动方程的整体解及相关问题的研究

We will consider the global-wellposedness and related problems of nonlinear Schrödinger and wave equation, in particular these two type equations related to Chern-Simons terms, such as abelian and non-abelian Chern-Simons-Schrödinger and Chern-Simons-Higgs system. These equations are used to describe some physics phenomenon such as Bose-Einstein condensation, refraction of light passing through a medium and quantum Hall effect in high temperature superconductivity, etc, so it will be of great significance to discuss the properties of these equations. We will use the tools from harmonic analysis, by establishing the multilinear estimates to consider the local-wellposedness of these equations. Further, combine with the conservation laws of the equations, e.g. mass and energy conservation, priori estimates and concentration compactness method to consider the global well-posedness of the equations. We will also use the heat flow method to choose the apropriate gauge potentials to consider the global well-posedness of the equations, which include the nonabelian gauge potentials. Combine with the conservation laws of the equations and analyse the parameters variation to describe the singularity formation in finite time. Establish the corresponding variational functional, and use the knowledge from critical point theory, such as mountain pass lemma, fountain theorem and implicit function theorem to consider the existence and multiplicity of the solitary solutions. We will also consider the mean-field limit and semiclassical limit problems of these two type equations.

我们会考虑非线性薛定谔方程和波动方程的整体解及相关问题，特别的与陈-西蒙斯项相关的非线性薛定谔方程和波动方程，像交换和非交换的陈-西蒙斯-薛定谔和陈-西蒙斯-希格斯方程组。这些方程被用来描述像波斯-爱因斯坦凝聚，光通过介质时的折射和高温超导的量子霍尔效应等物理现象，对于它们的性质的讨论是很有意义的。我们会运用调和分析的工具, 建立多线性估计来考虑方程的适定性。进一步的，结合方程的守恒律，像质量守恒和能量守恒，先验估计，集中紧性方法来考虑整体解的存在性。对于含有非交换的规范势的方程，我们会考虑运用热流的方法来选择合适的规范势来考虑方程的整体解的存在性。我们会结合方程的守恒律和分析相应参数的变化来考虑方程在有限时刻爆破的奇性刻划。 建立相应的变分泛函，运用临界点理论的知识，像山路引理，喷泉定理和隐函数定理来考虑孤波解的存在性和多解性。我们还会考虑平均场极限和半经典极限等问题。

我们考虑了非线性薛定谔方程和波动方程的整体解及相关问题，特别的与陈-西蒙斯项相关的非线性薛定谔方程和波动方程，像交换和非交换的陈-西蒙斯-薛定谔和陈-西蒙斯-希格斯方程组。这些方程被用来描述像波斯-爱因斯坦凝聚，光通过介质时的折射和高温超导的量子霍尔效应等物理现象，对于它们的性质的讨论是很有意义的。我们运用调和分析的工具, 建立多线性估计来考虑方程的适定性。进一步的，结合方程的守恒律，像质量守恒和能量守恒，先验估计，集中紧性方法来考虑整体解的存在性。对于含有非交换的规范势的方程，我们考虑运用热流的方法来选择合适的规范势来考虑方程的整体解的存在性。我们结合方程的守恒律和分析相应参数的变化来考虑方程在有限时刻爆破的奇性刻划。 建立相应的变分泛函，运用临界点理论的知识，像山路引理，喷泉定理和隐函数定理来考虑孤波解的存在性和多解性。我们还考虑了平均场极限和半经典极限等问题。我们证明了等仿的陈-西蒙斯-薛定谔在质量超临界的情形在散焦的情形解是整体存在并且是散射的，在聚焦的情形，解在小于基态解的时候是整体存在并且是散射的。