[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 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