# [M1 Seminar II] Week 5 : High Energy Estimates

Edited by Leun Kim

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 theorem, we can obtain the appropriate interval $[0,T]$ for this estimate. The main strategy is that we approximate $u_0$ by $u_0^k$ in $W^{s,2}$, prove the estimate for $u^k$, and then take the limit to obtain the estimate for $u$ using some inequalities and embeddings.

M1_Semi2_Week5

#### Leun Kim

Ph.D Candidate at The University of Tokyo
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.