流體力學

We will closely follow the online lecture notes by the famous mathematician Vladimar Sverak. Professor Sverak wrote his set of lecture notes while he taught his course on mathematical fluid dynamics during the year 2011 at the University of Minnesota. The main object of study will be incompressible Newtonian fluids, and the partial differential equations which we will discuss will be the very basic and elementary aspects of the incompressible Euler equation and the incompressible Navier-Stokes equation. Students taking this course should keep in mind that our course will definitely NOT cover any numerical aspect of mathematical fluid mechanics. The subject of computational fluid mechanics is totally beyond the scope of this elementary theoretical course (also far beyond the scope of expertise of the lecturer of this course). A student who is interested in the numerical aspect should really take a course in numerical analysis or mathematical modeling instead. Once we have discussed the elementary aspects of the course, there are two possible directions for further discussions: (First Direction): The theory of stationary Stokes equations and stationary Navier Stokes equation (we can discuss chapters 2, 3, 4 of the well-known text book : Lectures on Navier-Stokes Equations by professor Tai-Peng Tsai, Published by American Mathematical Society) (Second Direction) The relation between solution to stationary incompressible Euler equation and contact structure on a general Riemannian manifolds. (We can discuss Chapter 2 of the textbook "An Introduction to Contact Topology" by Hansjorg Geiges, published by Cambridge University Press). As such, we will try to first discuss the following important topics in mathematical fluid mechanics. Please note that the following list of topics are tentative, and there could be some changes of topics, dependent upon the time constraint and the atmosphere of the class. (Basic material to be covered) 1.Eulerian and Lagrangian description of fluid motion. 2. The equation of continuity. 3. The concept of vorticity. 4. Cauchy stress tensor in fluids. 5. Helmholtz-Hodge-Weyl decomposition in the framework of L^2 Lebesgue space. 6. The vorticity equation associated to the incompressible Navier-Stokes equation. 7. Kelvin's circulation theorem. 8. conservation of momentum, angular momentum, energy, and helicity of ideal incompressible fluids. 9. Hill's spherical vortex. 10. Tsai Tai Peng 's theorem about the non-existence of non-trivial self similar solution to the incompressible Navier-Stokes equation. 11. Stokes paradox for solutions to the stationary Navier-Stokes equations.

This course is for those first year graduate students, who have already passed the first course in Real Analysis (實變函數論). A student taking this course should have a solid understanding of basic undergraduate level mathematics, including linear algebra, vector calculus, the use of differential forms, and basic differential geometry.

Attendance: 25% Notes taking 25% Quiz 10% ONE Mid Term (in class) 15% Homework 10% Final Exam (in class) 15%

2011 lecture notes by Vladimar Sverak (available online in the personal web site of professor Sverak.) Other references: Lectures on Navier-Stokes Equations by professor Tai-Peng Tsai, Published by American Mathematical Society "An Introduction to Contact Topology" by Hansjorg Geiges, published by Cambridge University Press