From kinetic theory to fluid dynamics

The relationship between kinetic theory of gases and fluid dynamics has been one of important research topics in both mathematics and physics since the early 20th century. In this course, some aspects of this relationship are discussed mainly from the standpoint of physical and formal analysis. Specifically, we start with a summary of the Boltzmann equation and then explain the classical Chapman-Enskog and Hilbert expansions. The Chapman-Enskog expansion leads to the compressible Navier-Stokes equations. The slip boundary conditions for the compressible Navier-Stokes equations are also discussed. Furthermore, we show that, depending on the physical situation, fluid-dynamic equations of types other than traditional equations, such as the Euler and Navier-Stokes equations, can be derived from the Boltzmann equation. We mainly discuss the results based on formal analysis but also touch on rigorous mathematical results when necessary.

Theory and Application of Dispersive Wave Equations

In the first half of the 20th century, the theory of partial differential equations revolved around elliptic, hyperbolic and parabolic type equations.  In the second half of the century, a new type of equation appeared on the scene, namely dispersive equations.  Arising first in applications, these equations have taken a prominent place in the modern theory of partial differential equations.  At the same time, the number of applications of such equations has grown rapidly as well.
This set of lectures aims to introduce some of the modern mathematical theory that has emerged as well as survey a selection of the many real-world situations where dispersive equations arise as models for the phenomena in question.