Damped Euler-Poisson equations and nonlinear diffusion waves

阻尼欧拉-泊松方程和非线性扩散波

• 批准号: 354724-2011
• 负责人: Mei, Ming
• 金额: $ 0.73万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572546
• 关键词: Damped; Euler; Poisson; equations; nonlinear

Compressible flow through porous media with a dissipative external force field is usually described as a hyperbolic p-system with damping, a kind of Euler system of partial differential equations. The damping effect makes the system behave as a set of nonlinear diffusion equations, and the solutions possessing diffusion characters are known as nonlinear diffusion waves. Similar phenomena also occur in the hydrodynamic system for the models of semiconductor devices (Euler-Poisson equations). To study if the solutions of the original complex system behave as those diffusion waves and how to converge to the diffusion waves in the subsonic case, and to investigate the existence of the weak entropy solutions and their large-time behavior in the transonic case, the supersonic case, the case with vacuum, are very significant and challenging from both mathematical and physical points of view, and have been the hot research spots in applied partial differential equations The main purpose in this proposal is to investigate systematically the existence, uniqueness, of the solutions to the damped hydrodynamic Euler-Poisson system, and the construction of the solutions, the asymptotic behavior of the solutions to converge to nonlinear diffusion waves, stationary waves, and so on. The adopted methods will be the combination of the technical weighted energy method, the method of compensated compactness, the construction of entropic pairs, the method of shock wave structure. We are also interested in numerical computations to the large-time behavior of the solutions. We expect to develop some new techniques and methodologies to solve these long-time open questions, which will be a significant contribution to the theory of nonlinear partial differential equations and the application to fluid dynamics.

具有耗散外力场的多孔介质中的可压缩流动通常被描述为一个带阻尼的双曲p-系统，一种欧拉偏微分方程组。阻尼效应使系统表现为一组非线性扩散方程，具有扩散特征的解称为非线性扩散波。类似的现象也发生在半导体器件模型的流体动力学系统（欧拉-泊松方程）中。研究亚音速情况下原复杂系统的解是否表现为扩散波以及如何收敛到扩散波，以及研究跨音速、超音速和真空情况下弱熵解的存在性及其大时间行为，从数学和物理的角度来看都是非常有意义和具有挑战性的。一直是应用偏微分方程的研究热点 本文的主要目的是系统地研究有阻尼流体动力学Euler-Poisson方程组解的存在性、唯一性、解的构造、解收敛于非线性扩散波、驻波等的渐近性态，所采用的方法是技术加权能量法、补偿紧性法、熵对的构造，激波结构的方法。我们也对解的大时间行为的数值计算感兴趣。 我们期望能发展一些新的技术和方法来解决这些长期悬而未决的问题，这将对非线性偏微分方程理论及其在流体力学中的应用做出重大贡献。