The main subject of this course is the Navier-Stokes equations and the Euler equations. After introducing some fundamental facts of Sobolev spaces, we study basic notions of the Navier-Stokes and Euler equations. Finally, we study the latest results of the Navier-Stokes and Euler equations.
TheNavier-Stokes and Euler equations are fundamental equations in fluid dynamics, and these are applicable to wide variety of objects. On the other hand, these equations are not easy to comprehend without suitable training. To that end, rigorous proofs will be provided for most propositions, lemmas and theorems.
Students are expected to study the following:
・Understand Sobolev spaces.
・Be able to prove the local existence theorem of the Navier-Stokes equations.
・Be able to prove the local existence theorem of the Euler equations.
・Understand the illposedness theory of the Navier-Stokes equations.
・Understand the illposedness theory of the Euler equations
Schwartz distributions, Fourier transform, Sobolev spaces, the Navier-Stokes equations, Euler equations, commutator estimates
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Before coming to class, students should read the course schedule and check what topics will be covered.
Required learning should be completed outside of the classroom for preparation and review purposes.
Course schedule | Required learning | |
---|---|---|
Class 1 | Introduction to Fourier transforms and Schwartz class. | Details will be provided during each class session |
Class 2 | Properties of Fourier transform | |
Class 3 | distributional derivative, Fourier transform and convolution. | |
Class 4 | Convergence in distribution, examples of distributional Fourier transform | |
Class 5 | L^p function spaces, Hölder's inequality, Minkowski's inequality | |
Class 6 | Identification with distribution and L^p functions | |
Class 7 | Definition of Sobolev space and its properties | |
Class 8 | Introduction to the Navier-Stokes equations | |
Class 9 | Several estimates | |
Class 10 | Existence theorem of the Navier-Stokes equations | |
Class 11 | commutator estimate | |
Class 12 | A priori estimates for the Euler equations | |
Class 13 | Existence theorem of the Euler equations | |
Class 14 | Illposedness theory of the Navier-Stokes equations | |
Class 15 | Illposedness theory of the Euler equations |
None required
Fourier analysis and Functional analysis, Hitoshi Arai, Baifukan, ISBN 13 : 9784563006457 (Japanese)
Students' course scores are based on final exam and reports. Details will be provided in the class.
Students are expected to have passed Differential Equations I and Differential Equations II.