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

Edited by Leun Kim

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 high norms. And using that fact, we can prove that $u^k$ is Cauchy in the Banach space $C^0([0,T],L^2)$ so that $u^k \longrightarrow u$ in $C^0([0,T], L^2)$.

Then Gagliardo-Nirenberg’s inequality implies $u^k \longrightarrow u$ in $C^0([0,T],W^{s’,2})$ for $0 \leqslant s’ < s,$ $(s > 1+n/2)$ so that $u \in C^0 ([0,T], C_b^0)$ by the Sobolev embedding theorem. Finally we can conclude that $\partial_t u^{k}$ also converges in $C^0 ([0,T], W^{s-2,2}) \hookrightarrow C^0([0,T], L^2)$ so that $u \in C^1([0,T], L^2)$ and $u$ solves the system.

M1_Semi2_Week2

#### 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.