Dynamics of Wave Structures in Fluid Dynamics, Oscillatory Media, and Plasma Physics

流体动力学、振荡介质和等离子体物理中的波结构动力学

• 批准号: 1405728
• 负责人: Toan Nguyen
• 金额: $ 9万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405728&HistoricalAwards=false
• 关键词: Dynamics; Wave; Structures; Fluid; Oscillatory

The research will investigate the dynamics of boundary layers, coherent structures, and plasma equilibria. These objects are expressed as special solutions to mathematical models involving partial differential equations and have been widely used in various scientific disciplines. They are for instance of fundamental importance in biology, engineering, and physics. Utilizing boundary layer theory, engineers are able to significantly simplify the analysis of fluid flows near a solid body, such as a ship, or an airplane. This has proven exceptionally useful in aerodynamics. The PI will develop mathematical tools to study the validity of boundary layer simplifications of viscous flows, and thereby provide a deeper understanding of physical observations and laboratory experiments. The PI will also establish mathematical criteria, under which coherent structures in oscillatory media and equilibria of a plasma are well behaved under disturbances. The search for a stability criterion is important in practice, for instance, in helping engineers design stable devices, which could otherwise be damaged by unstable waves. The research focuses on mathematical questions concerned with the dynamics of boundary layers in fluid dynamics, the stability of coherent structures in oscillatory media, and the magnetic confinement of a plasma. The mathematical equations to be considered include the incompressible Euler and Navier-Stokes equations, general reaction-diffusion systems, and the relativistic Vlasov-Maxwell systems. Generic boundary layers of the Navier-Stokes equations are analytically shown to be spectrally unstable for sufficiently large Reynolds numbers. The PI will develop a nonlinear theory, building on the Fourier-Laplace transformed approach and the Evans function techniques, to prove the invalidity of boundary layer approximations in the vanishing viscosity limit. The research will also provide an understanding of the complete dynamics of nonlinear solutions near spectrally stable source defects. The PI will develop a novel nonlinear iteration scheme to study the stability properties of time-periodic traveling wave solutions, based on the spatial-dynamics techniques and the pointwise one-dimensional Green's function approach. Finally, the PI will initiate new investigations on magnetic mechanisms to confine a plasma modeled by the relativistic Vlasov-Maxwell systems.

该研究将研究边界层，相干结构和血浆平衡的动力学。这些对象表示为涉及部分微分方程的数学模型的特殊解决方案，并已广泛用于各种科学学科。它们在生物学，工程和物理学中具有根本重要性。利用边界层理论，工程师能够显着简化对固体体（例如船体或飞机）附近的流体流的分析。事实证明，这在空气动力学中非常有用。 PI将开发数学工具来研究粘性流的边界层简化的有效性，从而更深入地了解物理观察和实验室实验。 PI还将建立数学标准，根据该标准，在振荡培养基中的相干结构和等离子体的平衡在干扰下表现得很好。例如，在实践中搜索稳定标准很重要，例如，在帮助工程师设计稳定的设备时，否则这些设备可能会因不稳定的波浪而损坏。该研究的重点是与流体动力学中边界层的动力学有关的数学问题，振荡培养基中相干结构的稳定性以及等离子体的磁性约束。要考虑的数学方程包括不可压缩的Euler和Navier-Stokes方程，一般反应扩散系统以及相对论Vlasov-Maxwell Systems。在分析上，Navier-Stokes方程的通用边界层在足够大的雷诺数上在频谱上是不稳定的。 PI将发展一个非线性理论，建立在傅立叶式转换方法和Evans功能技术的基础上，以证明在消失的粘度极限中边界层近似的无效性。这项研究还将提供对光谱稳定源缺陷附近非线性溶液的完整动力学的理解。 PI将开发一种新型的非线性迭代方案，以研究基于空间动力学技术和尖锐的一维绿色功能方法，研究时间周期性波动波解决方案的稳定性。最后，PI将启动有关磁性机制的新研究，以局限以相对论Vlasov-Maxwell Systems建模的等离子体。

项目成果

Uniform in Time Lower Bound for Solutions to a Quantum Boltzmann Equation of Bosons

• DOI: 10.1007/s00205-018-1271-z
• 发表时间: 2016-05
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Toan T. Nguyen;Minh-Binh Tran

Sharp bounds for the resolvent of linearized Navier Stokes equations in the half space around a shear profile

剪切剖面周围半空间中线性纳维斯托克斯方程求解的锐界

• DOI: 10.1016/j.jde.2020.06.046
• 发表时间: 2020
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Grenier, Emmanuel;Nguyen, Toan T.
• 通讯作者: Nguyen, Toan T.

The Inviscid Limit of Navier–Stokes Equations for Analytic Data on the Half-Space

半空间解析数据纳维斯托克斯方程的无粘极限

• DOI: 10.1007/s00205-018-1266-9
• 发表时间: 2018
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: Nguyen, Toan T.;Nguyen, Trinh T.
• 通讯作者: Nguyen, Trinh T.