Boundary layers, Free boundaries and polymeric flows

MasmoudiDMS-1211806 本计画主要研究流体力学、气体动力学、电浆物理等问题。 主要研究者与他的合作者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决方案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波的整体存在性和散射。 他研究流体力学中的渐近问题，特别是那些引起边界层：零粘度极限和压缩-不可压缩极限的存在下，一个自由边界，以及流体动力学极限的玻尔兹曼方程在一个有界域。 了解边界层的性质在许多应用中是重要的。 他研究了均匀化椭圆算子中存在的振荡边界数据。 他还研究了一些非牛顿流体（粘弹性），特别是聚合物液体的弱解的整体存在性。 这些系统需要流体方程和聚合物的福克-普朗克方程之间的耦合。 研究者研究具有不同边界条件和不同极限情况的复杂系统的行为。 这些研究提供了一个更好的理解发生的真实的物理现象和域的有效性，每个简化模型。 这对于工程师和物理学家来说很重要，他们寻找最简单的模型来捕捉现象，以数字方式实现或应用于真实的生活中。 例如，水波问题对于理解海啸和巨浪很重要。 此外，粘弹性流体，特别是聚合物液体（例如白色，血液或面团）的研究在许多工业应用中很重要，如食品加工，并且在应用物理，化学和生物学的许多分支中具有很大的兴趣。 该项目包括培训数学和物理研究生。

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.