Boundary layers, Free boundaries and polymeric flows

MASMUDI DMS-1211806这个项目涉及对流体力学、气体动力学和等离子体物理问题的研究。首席研究人员与他的合作者P.Germain和J.Shatah开发了时空共振的方法，该方法将共振的标准概念与解的空间局部化研究相结合。该方法的一个应用是水波和毛细管波的整体存在和散射。他研究流体力学中的渐近问题，特别是那些引起边界层的问题：有自由边界时的零粘性极限和可压缩-不可压缩极限，以及有界域中Boltzmann方程的流体动力学极限。了解边界层的性质在许多应用中都很重要。他研究了存在振荡边界数据时椭圆算子的齐次化问题。他还研究了一些非牛顿流体(粘弹性)，特别是聚合物液体弱解的整体存在性。这些体系需要聚合物的流体方程和福克-普朗克方程之间的耦合。研究人员研究了具有不同边界条件的复杂系统在不同极限情况下的行为。这些研究提供了对发生的真实物理现象和每个简化模型的有效性领域的更好的理解。这对于寻找捕捉现象的最简单模型的工程师和物理学家来说很重要，无论是在数值上实现还是在现实生活中应用。例如，水波问题对于理解海啸和无赖海浪很重要。此外，粘弹性流体特别是聚合物液体(例如蛋清、血液或面团)的研究在食品加工等许多工业应用中都很重要，并且在应用物理、化学和生物学的许多分支中都引起了极大的兴趣。该项目包括对研究生进行数学和物理方面的培训。

MasmoudiDMS-1211806 This project is concerned with the study of problems coming from fluid mechanics, gas dynamics, and plasma physics. The principal investigator with his collaborators P. Germain and J. Shatah develops the method of space-time resonances, which combines the standard notion of resonances with the study of the spatial localization of solutions. One application of the method is the global existence and scattering for water waves and for capillary waves. He studies asymptotic problems in fluid mechanics, especially those giving rise to boundary layers: zero viscosity limit and compressible-incompressible limit in the presence of a free boundary, and the hydrodynamic limit of the Boltzmann equation in a bounded domain. Understanding the properties of the boundary layer is important in many applications. He investigates the homogenization of elliptic operators in the presence of oscillating boundary data. He also studies global existence of weak solutions for some non-Newtonian fluids (viscoelastic) and especially polymeric liquids. These systems require a coupling between a fluid equation and the Fokker-Planck equation for the polymers. The investigator studies the behavior of complicated systems with different boundary conditions and in different limiting cases. These studies provide an improved understanding of the real physical phenomena taking place and of the domain of validity of each simplifying model. This is important for engineers and physicists who look for the simplest model that captures the phenomena, to implement numerically or to apply in real life. For instance, water waves problems are important to understand tsunamis and rogue waves. Moreover, the study of viscoelastic fluids and especially polymeric liquids (egg white, blood, or dough for example) is important in many industrial applications such as food processing and is of great interest in many branches of applied physics, chemistry, and biology. This project includes the training of graduate students in mathematics and physics.