Well-Posedness and Singularity Formation in Applied Free Boundary Problems

应用自由边界问题中的适定性和奇异性形成

• 批准号: 2307638
• 负责人: David Ambrose
• 金额: $ 30万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307638&HistoricalAwards=false
• 关键词: Well; Posedness; Singularity; Formation; Applied

This project concerns the modelling of fluid motion as motivated by applications of dielectric fluids in microfluidic devices, the combustion and motion of flame fronts, and waves in water. The emphasis is in understanding for how long the mathematical models used to study such phenomena remain valid and predictive, in various physical regimes. For dielectric fluids, the focus is on situations where the density of electrical charge changes rapidly over small regions. For combustion and flame fronts, the aim is to study a hierarchy of mathematical models and determine how well the simpler models serve to approximate behavior in the more complicated models. While for water waves, the goal is to consider how wavetrains with different frequencies interact, or how wavetrains interact with certain kinds of bottom topography. The project provides research training opportunities for graduate students. The project will analyze the Melcher-Taylor leaky dielectric model, with the goal of establishing local well-posedness theory, study a mechanism for shock formation via analytical and numerical approaches, and consider global existence for the Kuramoto-Sivashinsky equation in more than one spatial dimension. Validation theorems relating the Kuramoto-Sivashinsky equation, coordinate-free models of flame fronts, and hydrodynamic flame models will be also investigated. Furthermore, the investigator will establish local well-posedness for water waves with spatially quasiperiodic initial data, and analyze related models, such as the Benjamin-Ono equation or the Euler equations for interfacial flows, in the spatially quasiperiodic setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目涉及介电流体在微流体设备中的应用，火焰锋的燃烧和运动​​以及水中的波动的启发。重点是理解用于研究这种现象在各种物理状态下使用的数学模型多长时间仍然有效和预测性。 对于介电流体，重点是电荷密度在小区域迅速变化的情况。 对于燃烧和火焰方面，目的是研究数学模型的层次结构，并确定更简单的模型在更复杂的模型中的近似行为。 对于水波而言，目标是考虑如何使用不同频率的波形相互作用，或者WaveTrains如何与某些类型的底部形状相互作用。该项目为研究生提供了研究培训机会。该项目将分析Melcher-Taylor泄漏的电介质模型，目的是建立局部适合度理论，研究通过分析和数值方法进行冲击形成的机制，并考虑在多个空间维度中为Kuramoto-Sivashinsky方程的全球存在。 还将研究与Kuramoto-Sivashinsky方程，无坐标模型和流体动力火焰模型有关的验证定理。 Furthermore, the investigator will establish local well-posedness for water waves with spatially quasiperiodic initial data, and analyze related models, such as the Benjamin-Ono equation or the Euler equations for interfacial flows, in the spatially quasiperiodic setting.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader影响审查标准。