Well-Posedness and Singularity Formation in Applied Free Boundary Problems

该项目涉及流体运动的建模，其动机是介电流体在微流体设备中的应用，火焰前锋的燃烧和运动以及水中的波浪。重点是了解用于研究这种现象的数学模型在各种物理状态下保持有效性和预测性的时间。 对于介电流体，重点是电荷密度在小区域内迅速变化的情况。 对于燃烧和火焰前锋，目的是研究数学模型的层次结构，并确定简单模型在更复杂的模型中的近似行为。 而对于水波，目标是考虑不同频率的波列如何相互作用，或者波列如何与某些类型的底部地形相互作用。该项目为研究生提供研究培训机会。该项目将分析Melcher-Taylor漏电介质模型，目标是建立局部适定性理论，通过分析和数值方法研究激波形成机制，并考虑Kuramoto-Sivashinsky方程在多个空间维度上的整体存在性。 验证定理有关的Kuramoto-Sivashinsky方程，坐标自由模型的火焰前锋，和流体动力学火焰模型也将进行研究。 此外，研究者还将利用空间准周期的初始数据建立水波的局部适定性，并分析空间准周期设置中的相关模型，如Benjamin-Ono方程或界面流的Euler方程。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.