Mathematical Structure in Fluid Mechanics

VasseurDms-0607953 研究者对流体力学和化学中作为模型出现的偏微分方程进行定性研究。 理解这些模型的性质是数学和科学中的一个基本挑战。 解的正则性和稳定性是与非平凡的物理现象如湍流和相变密切相关的问题，它们是复杂的多尺度问题。 分析这些现象的数学模型通常需要开发新的工具，包括分析，应用数学和随机学的技术。 这个项目，其中涉及理论和应用问题，从守恒定律到Navier-Stokes方程，动力学方程，以及流体中粒子的运动。 它需要非线性偏微分方程、渐近分析、概率论和随机论等理论的工具。 物理模型中数学结构的发现对于深入理解物理现象本身具有重要意义。 并对模型的有效性范围作了一些提示。 在这个项目中，该项目开发和研究涉及流体力学和远离平衡的物理学之间的耦合的新模型。 这一领域在科学和工程中有重要的应用。 例如，它模拟了河流中的污染物或发动机中汽油的燃烧。 应用和数学的挑战来自于现象的复杂性。

VasseurDms-0607953 The investigator carries out a qualitative study of partialdifferential equations arising as models in fluid mechanics andchemistry. Understanding the properties of these models is afundamental challenge both in mathematics and science. Theregularity and stability of solutions are issues strongly relatedto nontrivial physical phenomena like turbulence and phasetransitions that are complex multiscale problems. Analyzingmathematical models of these phenomena often requires thedevelopment of novel tools encompassing techniques from analysis,applied mathematics, and stochastics. This project, whichaddresses both theoretical and applied issues, spans fromconservation laws to Navier-Stokes equations, kinetic equations,and motion of particles in fluids. It requires tools from thetheories of nonlinear partial differential equations, asymptoticanalysis, probability, and stochastics. The discovery of mathematical structure in physical modelsis of great importance for a deeper understanding of the physicalphenomena themselves. It gives also some hints of the range ofvalidity of the models. In this project, the investigatordevelops and studies new models involving couplings between fluidmechanics and physics far from equilibrium. This field hasimportant applications in science and engineering. For example,it models pollutants in a river or combustion of gasoline in anengine. The applied and mathematical challenges stem from thecomplexity of the phenomena.