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 II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

We proved the right-continuity of the solution at the initial time in $W^{s,2}$ which implies the continuity of the whole interval. And by the PDE and Sobolev embedding theorem, we can easily get the regularity in the existence theorem. Also … Continue reading

[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 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