# [M1 Seminar II] Week 4 : Regularities of the Solution and and Improved Existence Theorem

Edited by Leun Kim

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 we can improve the theorem a little bit more. The theorem can be improved with respect to the length $T$ of the interval of existence, that is, the dependence on $\| u_0 \|_{s,2}, g_2$ can be weakened.

M1_Semi2_Week4

#### 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.