随机非线性薛定谔方程几类问题的研究

Stochastic nonlinear Schrödinger equation is a new type of stochastic partial differential equation with the dispersive structure, and it plays fundamental roles in the study of quantum physics as well as in nonlinear dispersive waves, thereby being of valuable importance in both applications and mathematics. Nowadays, the study of nonlinear Schrödinger equations mainly concentrates on two areas: the blow-up mechanics in the focusing mass-critical case and the invariant measure associated with random data. Representative achievements include respectively the profile decomposition methods developed by Kenig and Merle and Bourgain’s investigation of the influences of invariant measure on the global well-posedness. On the other hand, in the field of stochastic partial differential equations, two important research topics, with valuable probability significance, are the investigations of noise effects on deterministic systems and the properties of invariant measures. Therefore, for the stochastic nonlinear Schrödinger equations driven by noises, the applicant wishes to study the fundamental properties of blow-up solutions and invariant measures, including the regularizations of noise and invariant measure on blow-up and global well-posedness respectively... In addition, the logarithmic Schrödinger equation is another important nonlinear equation, with widely applications in various fields such as nuclear physics, photochemistry and so on. Taking into account the stochastic perturbations in real applications, the applicant wishes also to study the global well-posedness of stochastic logarithmic Schrödinger equation.

随机非线性薛定谔方程是一类新型的具有色散型结构的随机偏微分方程，是量子物理和非线性色散波研究中的一类基本方程，具有重要的应用价值和数学意义。目前国际上非线性薛定谔方程非常活跃的两个研究方向是：聚焦质量临界情形下的爆破机制以及随机初值下方程的不变测度；代表性的进展分别有，Kenig和Merle的波包分解方法，以及Bourgain的不变测度与全局适定性的联系。另一方面，在随机偏微分方程领域中，噪声对确定系统的影响以及不变测度的性质是十分重要的研究课题，具有重要的概率意义。因此，对于含噪声驱动的随机非线性薛定谔方程，申请人希望研究爆破解和不变测度的基本性质，包括噪声对爆破的正则化效果，以及不变测度对全局适定性的正则化效果。.此外，对数薛定谔方程是另一类重要的非线性方程，在核物理和光化学等诸多领域有广泛的应用。考虑到在实际应用中往往有噪声的干扰，申请人还希望研究随机对数薛定谔方程的全局适定性。

随机非线性薛定谔方程是一类新型的具有色散型结构的随机偏微分方程，是量子物理和非线性色散波研究中的一类基本方程，具有重要的数学意义和应用价值。本项目主要研究随机非线性薛定谔方程的适定性、长时间行为、爆破解以及最优控制等几类重要问题。目前已取得的科研成果包括有：次临界情形下该类方程的最优控制问题以及长时间散射行为，临界情形下的全局适定性、散射和Stroock-Varadhan型支集定理，（超）临界情形下噪声对爆破解的正则化效果，含对数非线性项的随机薛定谔方程的全局适应性，以及更一般的色散方程的Strichartz估计和局部光滑估计等。这些结果给出了随机非线性薛定谔方程解的适定性和长时间行为较为完整的刻画，并为更一般的随机色散型方程展开研究奠定了基础。