#### Local well-posedness of the fifth-order KdV equations for rough data using short time $X^{s,b}$ structures = $X^{s,b}$ 구조를 이용한 5계 KdV 방정식의 해의 존재성 연구

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In this paper, we mainly prove that the following the fifth-order equation arising from the Korteweg-de Vries (KdV) hierarchy \begin{equation*} \begin{cases} \pt u + \px^5 u + c_1\px u\px^2 u + c_2u\px^3 u = 0 \qquad x,t \in \R \\ u(0,x) = u_0(x) \qquad u_0 \in H^s(\R) \end{cases} \end{equation*} is locall well-posed with initial data in $H^s(\R)$ for $s > \frac54$. \\ The method is a short time $X^{s,b}$ space, which is developed by Ionescu-Kenig-Tataru \cite{IKT} in the context of the KP-I equation. In addition, we use a weight on $X^{s,b}$ to reduce the contribution of high-low frequency interaction where the low frequency has large modulation in the proof of energy estimate. As an immediate result from a conservation law, we have the second equation in the KdV hierarchy, $$\partial_t u - \px^5 u -30u^2\px u + 20\px u\px^2 u + 10u\px^3u=0$$ is globally well-posed in $H^2$.\\ Moreover, we introduce the standard \emph{$X^{s,b}$ space} and counter-examples that the nonlinear estimates fails in the usual \emph{$X^{s,b}$ spaces}. We also prove that the KdV equation is locally well-posed with initial data in $H^s(\R)$ for $s > -\frac34$ by using the Picard iteration argument.
Kwon, Soon-Sikresearcher권순식
Description
한국과학기술원 : 수리과학과,
Publisher
한국과학기술원
Issue Date
2012
Identifier
509385/325007  / 020104246
Language
eng
Description

학위논문(석사) - 한국과학기술원 : 수리과학과, 2012.8, [ ii, 54 p. ]

Keywords

$X^{s; b}$ space; local well-posedness; the KdV equation; $X^{s; b}$ 함수 공간; 해의 존재성; KdV 방정식; 5계 KdV 방정식; the fifth-order KdV

URI
http://hdl.handle.net/10203/181587