量子等离子体中的流体动力学方程与色散波方程的相关数学研究

This project is dedicated to the related mathematical problems of the quantum hydrodynamic models for quantum plasmas, which include the quantum Navier-Stokes equations, the quantum ion acoustic equations, the quantum Korteweg-de Vries equations, the quantum magnetohydrodynamic model, the quantum hydrodynamics model for semiconductor devices, the quantum Zakharov system, and the quantum Schrödinger-Poisson type system. We first study the local and global well-posedness, the existence of finite energy weak solution and global classical solutions, turbulence theory, singular limit and Rayleigh-Taylor instability for the first five quantum systems. We next establish the strict mathematical derivation, well-posedness of global smooth solution and singular limit for the quantum Zakharov system. Finally, we analyze the crystal ground state, semiclassical limit, local and global well-posedness and regularity for the quantum dispersive equations and the relative systems such as the quantum Schrödinger-Poisson system and the quantum Schrödinger-Poisson-Slater system.

研究量子等离子体中的量子流体动力学方程:量子Navier-Stokes方程,量子离子声波方程,量子Korteweg-de Vries方程,量子磁流体动力学方程,半导体量子流体动力学方程,量子Zakharov系统及量子Schrödinger-Poisson型系统.首先研究前五类量子流体系统解的局部,整体适定性,有限能量弱解的存在性,整体经典解的存在性,湍流理论,奇性极限及Rayleigh-Taylor不稳定性.其次研究量子Zakharov系统的严格数学导出,其整体光滑解的适定性及奇性极限.最后研究量子色散波方程及相关系统,如Schrödinger-Poisson系统,量子Schrödinger-Poisson-Slater系统晶体基态,半经典极限,局部,整体适定性及正则性.

在本项目的执行过程中，我们得到了比较丰富的数学结果，研究工作取得了一定的进展, 主要研究非局部非线性Schrodinger方程的驻波及整体解的存在性、具有分数阶拉普拉斯算子粘性的Camassa-Holm方程的大时间行为及收敛性、非局部非线性Schrodinger方程解整体存在的最佳条件问题、具有分数阶扩散的Camassa-Holm方程解的整体存在性及正则性及高阶Benjamin-Ono方程柯西问题的局部适定性。 这些结果均发表于国际国内权威学术期刊上： Nonlinearity 、Calc. Var. Partial Differential Equations、Discrete and Continuous Dynamical Systems、Chin. Ann. Math. Ser. B 、Forum Math.。