Local well-posedness and regularity properties for an initial-boundary value problem associated to the fifth order Korteweg-de Vries equation

Abstract

In this work we prove that the initial-boundary value problem (IBVP) for the fifth order Korteweg-de Vries equation \begin{align*} \left. \begin{array}{rlr} u_t+\partial_x^5 u+u\partial_x u&\hspace{-2mm}=0,&\quad x\in\mathbb R^+,\; t\in\mathbb R^+,\\ u(x,0)&\hspace{-2mm}=g(x),&\\ u(0,t)=h_1(t),\, \partial_x u(0,t)&\hspace{-2mm}=h_2(t),\,\partial_x^2 u(0,t)=h_3(t), \end{array} \right\} \end{align*} is locally well posed, when the data $g$, $h_1$, $h_2$, $h_3$ are taken in such a way that $g\in H^s(\mathbb R_x^+)$, and $h_{j+1}\in H^{\frac{s+2-j}5}(\mathbb R_t^+)$, $j=0,1,2$, $s\in [0,\frac{11}4)\setminus \{\frac12,\frac32,\frac52\}$, and satisfy the following compatibility conditions: \begin{align*} g(0)=h_1(0) \text{ if } \frac12<s<\frac32;\\ g(0)=h_1(0),\; g'(0)=h_2(0) \text{ if } \frac32<s<\frac52;\\ g(0)=h_1(0), \; g'(0)=h_2(0),\; g''(0)=h_3(0) \text{ if } \frac52<s<\frac{11}4. \end{align*} Besides, we prove that the nonlinear part of the solution is smoother than the initial datum $g$.