2019 Advanced topics in Analysis E1

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Academic unit or major
Graduate major in Mathematics
Kagei Yoshiyuki 
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Course description and aims

This course gives the theory of bifurcation and stability for the compressible Navier-Stokes equations which are known to be the fundamental equations in the fluid mechanics. The compressible Navier-Stokes equations are classified in a class of quasilinear heyperbolic-parabolic systems and have provided fundamental issues in the field of partial differential equations such as existence, uniqueness, regularity and asymptotic behavior of solutions, and etc. In this course, the standard bifurcation theory for partial differential equations is firstly explained; and an application of this theory is illustrated by considering stationary bifurcation problems for the incompressilble Navier-Stokes equations which is classified in a class of parabolic systems. Secondly, stationary bifurcation problems for compressilble Navier-Stokes equations is considered, which cannot be treated by the standard bifurcation theory. This course will be completed with "Advanced topics in Analysis F" in the next quarter.

The aim of this course is to learn some aspects of mathematical analysis of nonlinear partial differential equations through the bifurcation and stability analysis of the Navier-Stokes equations.

Student learning outcomes

・Understand differentiaion and implicit function theorem for nonlinear maps in infinite dimensional spaces.
・Understand the standard bifurcation theory.
・Understand resolvent and spectrum of linear operator.
・Understand fundamental properties of nonlinear parabolic equtiations.


Frechet derivatice, implicit function theorem, resolvent, spectrum, Lyapunov-Schmidt method, bifurcation analysis, incompressible Navier-Stokes equations

Competencies that will be developed

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

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  Course schedule Required learning
Class 1 The following topics will be covered in this order : -- Differentiation of nonlinear maps in infinite dimensional spaces: Frechet derivatives, Taylor expansions, etc. -- Contraction mapping principle -- Implicit function theorem -- Standard bifurcation theory -- Stability of bifurcating solutions -- Bifurcation and Stability analysis of the incompressible Navier-Stokes equations Details will be provided during each class session.


None required

Reference books, course materials, etc.

Details will be provided during each class.

Assessment criteria and methods

Attendance and Assignments.

Related courses

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

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


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