General boundary value problems of a class of fifth order KdV equations on a bounded interval

Abstract

This paper investigates the initial boundary value problem (IBVP) associated with the fifth-order Korteweg-de Vries (KdV) equation on a finite interval: \begin{equation} \begin{cases} {\partial _t} u+{\partial _x} u+\beta {\partial _x} ^3u +{\partial _x} ^5u= G(u), \ \ 0 < x < L , \\ u(x,0)=\phi(x), \end{cases}\tag{0.1} \end{equation} subject to non-homogeneous boundary conditions: \begin{equation} B_j u = h_j(t), \quad j = 1,2,3,4,5, \quad t > 0, \tag{0.2} \end{equation} where $ G(u)= c_0 u{\partial _x} u + c_1{\partial _x} u{\partial _x}^2 u + c_2 {\partial _x} (u{\partial _x}^2 u) + c_3 {\partial _x} (u^3), $ and $$ B_j u = \sum_{k=0}^{4} (a_{jk} \partial_{x}^{k}u(0,t) + b_{jk} \partial_{x}^{k}u(L,t)), \ j=1, \cdots, 5, $$ and $a_{jk}, b_{jk}$ are real constants for $k=0,1,2,3,4$ and $j=1,2,3,4,5$. Under certain general assumptions on these coefficients, the well-posedness of the IBVP $(0.1)-(0.2)$ is established, showing local analytical well-posedness in the space $H^{s}(0,L)$ for $s \geq 1$ (and for some specified cases, $s \geq 0$), with initial data $\phi \in H^{s}(0,L)$ and boundary values $h_j$ belonging to appropriate spaces with optimal regularity. In contrast, for the pure initial value problem (IVP) of the fifth-order KdV equation posed on $\mathbb R$, it is known to be continuous ($C^0$) well-posed in the space $H^s (\mathbb{R})$ for $s \geq 2$, but it fails to be analytically well-posed in $H^s (\mathbb{R})$ for any $s \in \mathbb{R}$. The remarkable strong Kato smoothing and double sharp Kato smoothing properties, attributed to Kenig, Ponce, and Vega [42, 45], of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line $\mathbb{R}$, $$ {\partial _t} v +\beta {\partial _x}^3 v + {\partial _x} ^5 v = g (x,t), \quad v(x,0) =0, \ x, \ t \, \in \mathbb{R}, $$ have played a pivotal role in establishing the well-posedness of the IBVP $(0.1)-(0.2)$.