[M1 Seminar II] Week 3 : Regularities of the Solution

This week we will prove that the solution of the system $u$ is in $C^0([0,T], W^{s,2}) \cap C^1([0,T], W^{s-1,2})$ so that $u \in C_b^1 ([0,T] \times \Bbb R^n)$. For this purpose we first show that $u \in L^\infty ([0,T], W^{s,2})$, … Continue reading

[M1 Seminar II] Week 2 : Local Existence for Quasi-linear Symmetric Hyperbolic Systems (2)

This week, we will prove the local existence of quasi-linear symmetric hyperbolic systems by using $u^{k}$ which is iteratively defined by the solution of the linear symmetric hyperbolic system. For this purpose we first proved the boundedness of $u^k$ in … Continue reading

[M1 Seminar] Week 13 : Elementary Inequalities on the Klainerman Vector Field

We construct the Klainerman vector field with some operators which were defined in the last seminar. And we prove some basic inequalities of these family of operators which play an important role for our final goal. M1_Semi_Week13  

[M1 Seminar] Week 11,12 : Introduction to the Klainerman Vector Field

We introduce the Klainerman vector field and prove some commutator relations which will play an important role in the proof of the global existence theorem for the nonlinear wave equations. M1_Semi_Week11-12  

[M1 Seminar] Week 6,7 : A Global Existence Theorem to the Linear Symmetric Hyperbolic Systems

This week, we prove the global existence theorem to the linear symmetric hyperbolic system by using the standard energy inequality. M1_Semi_Week6-7  

[M1 Seminar] Week 5 : Energy Estimates for the Linear Symmetric Hyperbolic System

We derive the corresponding estimates for higher derivatives from the basic energy estimates which was given in the last week. We also prove a simple corollary using this result. And we discuss about the finite propagation speed of the hyperbolic … Continue reading

[M1 Seminar] Week 2 : Decay Estimates for Linear Wave Equations

   This week, we have considered only odd space dimensions for the linear wave equation to prove two estimates in Theorem 2.1. The proof was done for the simple case. For the general cases including the case of even dimensions, we … Continue reading

[M1 Seminar] Week 1 : Introduction to Nonlinear Wave Equations

   For nonlinear wave equations, we cannot guarantee that the solutions exist for some initial value problems. Also the global solutions may exist but also it may have some singularities depending on the initial conditions. Thus we need the global existence … Continue reading

서강대 졸업 프로젝트

    여름 방학 때부터 끌어오던 졸업프로젝트를 드디어 내일 발표회를 끝으로 마무리 짓게 되었습니다. 주제는 Finite Element Method(유한요소법)을 통한 전자기적 문제 해석입니다. FEM은 PDE를 푸는 수치해석적 방법 중 하나이며 최근 실로 많은 분야에 그 쓰임새가 점차 많아지고 있는 유용한 방법입니다.     … Continue reading