Well-Posedness and Singularity Formation in Applied Free Boundary Problems

该项目涉及微流控装置中应用介电流体、火焰锋面的燃烧和运动以及水中波的流体运动的建模。重点是理解用于研究这种现象的数学模型在不同的物理制度下保持有效和预测的时间。对于介电流体，重点关注电荷密度在小区域内快速变化的情况。对于燃烧和火焰前锋，目的是研究数学模型的层次结构，并确定较简单的模型在较复杂的模型中对行为的近似程度。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海底地形相互作用。该项目为研究生提供了研究培训机会。该项目将分析Melcher-Taylor泄漏介质模型，目的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的机制，并在多个空间维度上考虑Kuramoto-Sivashinsky方程的整体存在性。与Kuramoto-Sivashinsky方程、火焰锋面无坐标模型和流体动力火焰模型相关的验证定理也将被研究。此外，研究人员将利用空间准周期初始数据建立水波的局部适定性，并分析相关模型，如空间准周期环境中的本杰明-小野方程或界面流动的欧拉方程。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.