1차원 클라인 골든 방정식에 대한 고찰 (Time Decay 의 관점에서)

Klein-Gordon-Equation

이 쯤 해서 내 나름대로 정리도 할 겸 클라인 골든 방정식(Klein-Gordon Equation)에 대한 포스트를 써 두어 본다. 먼저 1절에서는 $n$ 차원 클라인 골든 방정식에 대한 고전 결과들을 소개하고, 2절에서는 업계(?)에서 주로 쓰는 용어들을 설명한다. 3절에서는 단독(単独) 1차원 클라인 골든 방정식에 … Continue reading

History of Some Major Works to the Klein-Gordon equations

I15-81-Dirac

Nonlinear Klein-Gordon Equation(NLKG) 의 역사(?)를 간단히 정리해 보았습니다. 방정식 자체의 역사는 길지 몰라도, 방정식을 “수학적인 방법”으로 공략해서 성과를 얻어낸 역사는 생각보다 그리 길지 않습니다. 구체적으로 다음과 같은 형태의 NLKG의 역사에 대해 알아보겠습니다. $$(\square +1) u = F(u, \partial u),\;\;\;t\geqslant 0, … Continue reading

[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem which was proven before gives $n >5$ for the quadratic nonlinearities. But in fact this … Continue reading

[M1 Seminar II] Week 7 : A Global Existence Theorem for Nonlinear Wave Equations

We consider the following initial value problem \[ y_{tt} – \Delta y = f(Dy,\nabla Dy)\;\;\;\;\text{with}\;\;\;\; y(t=0) = y_0,\; y_t (t=0) = y_1 \tag{P} \] with $f \in C^\infty ( \Bbb R^{(n+1)^2}, \Bbb R)$, $\exists \alpha \in \Bbb N$ such that … Continue reading

[M1 Seminar II] Week 6 : Weighted a priori Estimates for Small Data

For the proof of the global existence theorem of nonlinear wave equations, this a priori estimates is essential as well as the high energy estimates. Also the high energy estimates play an important role in the proof of a priori … Continue reading

[M1 Seminar II] Week 5 : High Energy Estimates

We prove the high energy estimates for the nonlinear wave equation on the nonlinearity $\alpha =1$. In fact, this problem is a special case of the quasi-linear symmetric hyperbolic system with some assumptions. By the help of the previous existence … Continue reading

[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 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