2016 Special courses on advanced topics in Mathematics E

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Yanagida Eiji  Yoneda Tsuyoshi 
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Day/Period(Room No.)
Intensive ()  
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Syllabus updated
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

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

  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

Reference books, course materials, etc.

Fourier analysis and Functional analysis, Hitoshi Arai, Baifukan, ISBN 13 : 9784563006457 (Japanese)

Assessment criteria and methods

Students' course scores are based on final exam and reports. Details will be provided in the class.

Related courses

  • MTH.C341 : Differential Equations I
  • MTH.C342 : Differential Equations II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Differential Equations I and Differential Equations II.

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