2020 Special lectures on advanced topics in Mathematics E

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Academic unit or major
Graduate major in Mathematics
Kawashima Shuichi  Kagei Yoshiyuki 
Course component(s)
Mode of instruction
Day/Period(Room No.)
Intensive (Zoom)  
Course number
Academic year
Offered quarter
Syllabus updated
Lecture notes updated
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Course description and aims

This course gives lecture on mathematical analysis for hyperbolic systems of balance laws. We develope the general mathematical theory on the global existence and asymptotic stability of solutions.

The aim of this course is as follows. We treat hyperbolic systems of balance laws as nonlinear partial differential equations. We formulate structural conditions on the system and under these structural conditions, we prove the global existence and asymptotic stability of solutions for small initial data. The proof is based on the energy method and the semigroup technique.

Student learning outcomes

・To understand the physical and mathematical meaning of the structural conditions
・To understand the dissipative structure of the system
・To learn the energy method
・To learn the semigroup technique
・To understand the mathematical theory on the global existence and asymptotic stability of solutions


hyperbolic system of balance laws, mathematical entropy, symmetrization of system, stability condition, craftsmanship condition, dissipative structure, decay property, global solution, energy method, a priori estimate, asymptotic stability, semigroup technique

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 ・Introduction (Hyperbolic system of balance laws) ・Mathematical entropy and symmetrization ・Decay property for linearized system ・Dissipative structure ・Results on the global existence ・Energy estimates (A priori estimates) ・Time-weighted energy estimates ・Decay estimates Details will be provided during each class session


None required.

Reference books, course materials, etc.

Reference texts will be given during the class.

Assessment criteria and methods

Assignments (100%).

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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