• On some boundary problems of the Boltzmann equation and its hydrodynamic limit by Prof. Chanwoo Kim
In this lecture series we will learn about (1) Basic properties of the Boltzmann equation; (2) Boundary problems; (3) L^2 coercivity estimate; (4) L^\infty estimate: L^p-L^\infty Bootstrap argument; (5) Global solvability and Stability of some boundary problems of the Boltzmann; (6) Hydrodynamic limit of the Boltzmann equation.
— References:
- Glassey’s book of the Cauchy problem of kinetic theory
- Guo: Bounded solutions for the Boltzmann equation, Quarterly of Applied Mathematics Vol. 68, No. 1 (March 2010), pp. 143-148
- Esposito, Guo, Kim, Marra: Stationary solutions to the Boltzmann equation in the Hydrodynamic limit (Arxiv: 1502.05324v3)
Slides are available at here.
• Recent results in the mathematical analysis of inviscid flows by Prof. Ondrej Kreml
In this course we will study mainly the incompressible Euler system of partial differential equations describing the flows of ideal inviscid fluid. We will start with stating the basic properties of the system and show the difference between the 2D and the 3D cases with respect to existence theorems for smooth enough solutions. The main part of the course will cover the convex integration technique and its application to the incompressible Euler system. For the historic context we will start with the ideas of the proof of theorem of Nash and Kuiper for isometric embeddings, a seemingly nonrelated problem which was solved by a similar technique. Then we will continue with proving the existence of compactly (in space-time) supported weak solutions to the incompressible Euler system with detailed proofs of the key aspects. We will then summarize the results achieved by application the technique and cover the story leading to the proof of the famous Onsager’s conjecture for incompressible Euler system. Finally, we will show some of the applications of this theory to the compressible isentropic Euler equations.
— References:
- Book “Mathematical Theory of Incompressible Nonviscous Fluids” by C. Marchioro and M. Pulvirenti (for derivation of the incompressible model and properties of classical solutions)
- Papers “The Euler equations as a differential inclusion”, “On Admissibility Criteria for Weak Solutions of the Euler equations” and Survey “The h-principle and the equations of fluid dynamics” by C. De Lellis and L. Szekelyhidi (all three for the main results concerning nonuniqueness of weak solutions to the incompressible Euler equations).
- Lecture notes “From Isometric Embeddings to Turbulence” by L. Szekelyhidi (for general framework and context of the theory of De Lellis and Szekelyhidi)
• An introduction to instabilities in interfacial fluid mechanics by Prof. Ian Tice
Free boundary problems in fluid mechanics occur when a fluid interacts with another fluid or a moving solid, and the interface between these different phases is free to evolve in time. These problems are ubiquitous in nature and occur in an enormous range of scales: blood flow in elastic arteries, the floating of oil slicks on the ocean, and the interface between solar plasma and the vacuum of space are all examples. The main purpose of these lectures is to introduce some classical mathematical techniques for studying interfacial instabilities and to explore some important examples, such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Time permitting, we will also explore some modern extensions of these techniques.
— References:
- S. Chandrasekhar – Hydrodynamic and Hydromagnetic Stability