Zakharov系统的解的动力学行为研究

Firstly, this research will consider the global dynamics slightly above the ground state energy for the 3D Zakharov system in radial case. All the results obtained before are for the case with energy below the ground state. The main ingredient of this research is to construct a center stable manifold in a small neighborhood of ground state; and then this research will give a complete description of the evolution in this neighborhood by variational estimates...Secondly, this research will consider the global well-posedness and scattering for the 3D vector Zakharov system with small energy. Since radial symmetry cannot be keep by this system, the improved radial Strichartz estimates, which has been used for radial case, can not be used directly. We will try to choose proper work space and generalize improved radial Strichartz estimates to this space; and then to prove the global well-posedness and scattering of this system by contracting mapping principle. ..This research will provide necessary theory evidence for analyzing the evolution law of wave function, and will be helpful to explain and predict the related physical phenomena. It has important theory and research value.

首先，本课题将在能量空间中研究三维径向情形下Zakharov系统在系统能量稍高于基态能量时解的长时间动力学行为．在此之前的研究结果都是针对基态能量以下的情形．本课题拟结合基态附近的双曲结构构造中心稳定流形，再结合变分法将基态能量附近解的动力学行为按散射和破裂进行分类．..其次，本课题将研究三维向量Zakharov系统在小初值条件下的适定性和散射．由于这一系统不保持径向，现有的处理径向情形所用到的改进的径向Strichartz估计不能直接应用，本课题拟选取合适的工作空间，并将改进的径向Strichartz估计推广到这一空间，进而应用压缩映射原理证明这一系统在小初值时解的适定性和散射．..本研究将为分析波函数随时间的演变规律提供理论依据，可用于解释和预测相关物理现象，具有重要的理论意义和研究价值．