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.

- Advisors
- Kwon, Soon-Sik
*researcher*; 권순식

- 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

- Appears in Collection
- MA-Theses_Master(석사논문)

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