流体动力学模型解的存在性及稳定性研究

The properties of solutions to shallow water equations and Euler-Poisson equations are active subjects in the research of fluid mechanics. The classical solutions to shallow water equations are under consideration in order to desctribe the horizontal structure of the fluid (such as tsunami, dam break and so on). However, it becomes more difficult and complicated when investigating classical solutions due to the strong degeneracy near vacuum region and the arbitrariness of exteral force. On the other hand, Euler-Poisson equations models several important physical flows including the propagation of electrons in plasmsa and so on. As a first step to understand the model, many researchers involved in the work on some important physically relevant wave patterns including transonic flows and so on. The applicant and her cooperators have obtained the local well-posedness of Cauchy problem for shallow water equations containing vacuum and the existence of steady subsonic Euler flow in three dimensional nozzle. Based on the work above, this programm intends to 1)investigate the Cauchy problem for shallow water equations containing vacuum, and in the case of constant viscosity coefficients, obtain a blow-up phenomenon for compactly supported initial height, as well as global well-posedness for nonvacuum initial far fields; and in the case of density-dependent viscosity coefficients，obtain global well-posedness for spherically symmetric initial data; 2)obtain the existence and dynamical stability of steady transonic shock solutions for compressible Euler-Poisson equations in both divergent and convergent qusi-one-dimensional nozzle with suitable different electric fields.

关于浅水波方程组和欧拉-泊松方程组解的研究是流体力学研究中十分活跃的课题。人们对流体横向结构(如海啸、溃坝问题等)的了解基于对浅水波方程组光滑解的研究，而真空区域附近的强退化现象和外力的任意性使得该研究变得更加困难和复杂。另一方面，欧拉-泊松方程组描述了一类重要的物理流，如等离子体中电子的传播等。与该模型相关的一些具有重要物理意义的波形（如跨音速激波等）的研究一直受到广泛关注。申请人及其合作者已得到含真空浅水波方程组柯西问题的局部适定性以及三维管道中稳态亚音速欧拉流的存在性。在此基础上，本项目1）研究含真空的浅水波方程柯西问题：粘性系数为常数时，当初始深度具紧支集时得到爆破现象,当初始深度在无穷远处远离真空时得到整体适定性；粘性系数依赖于深度时，得到球对称初值下整体适定性；2）研究欧拉-泊松方程在拟一维开口和闭口管道中稳态跨音速激波解的存在性,并分别在不同电场条件下，得到该解的动态稳定性。

受项目资助以来, 申请人及其合作者推导了新的含浅水波方程的生物趋向性模型，用以研究生物数学中的生物趋向性-Navier-Stokes模型，同时建立了强解的局部存在性和解的爆破准则；其次，我们证明了在一定物理边界条件下，在拟一维管道中欧拉-泊松系统的跨音速激波解在开口和闭口管道中都是动态稳定的，关键是建立估计用以描述电场和管道几何形状的相互影响；同时，我们研究了旋度对于无穷长管道中非等熵光滑亚音速欧拉流的影响，证明了大旋度假设下，欧拉方程的光滑亚音速解的存在唯一性；此外，我们研究了一维Navier–Stokes–Korteweg 方程的粘性消失极限问题，当相应的欧拉方程有稀疏波解时，如果忽略初始层或相应稀疏波解是光滑的，我们证明了Navier–Stokes–Korteweg 方程具有中心稀疏波的解存在，并在粘性系数或毛细作用消失时收敛到相应欧拉方程的稀疏波解。