### 2020　Special lectures on current topics in Mathematics K

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Instructor(s)
Kawashima Shuichi
Course component(s)
Lecture
Day/Period(Room No.)
-
Group
-
Course number
MTH.E641
Credits
2
2020
Offered quarter
3Q
Syllabus updated
2020/3/24
Lecture notes updated
-
Language used
English
Access Index ### 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

### Keywords

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

 Intercultural skills Communication skills ✔ Specialist 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. 