# [M1 Seminar II] Week 3 : Regularities of the Solution

Edited by Leun Kim

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 $u \in L^\infty ([0,T], W^{s,2})$, and using this fact, we can show that $u$ is weakly continuous from $[0,T]$ into $W^{s,2}$, and also $u$ is Lipschitz continuous from $[0,T]$ into $W^{s-1,2}$ by the Alaoglu’s theorem and the Riesz representation theorem.

To prove $u \in C^0([0,T], W^{s,2}) \cap C^1 ([0,T], W^{s-1,2})$, in fact, we are sufficient to show the right-continuity of $u$ at $t=0$ because the right-continuity of $u$ at $0$ implies the right-continuity on $[0,T)$, and the right-continuity on $[0,T)$ implies the left-continuity on $(0,T]$. Also by the PDE, $u \in C^0 ([0,T], W^{s,2})$ implies $u \in C^1([0,T], W^{s-1,2})$.

M1_Semi2_Week3

#### 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.