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

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 $h_{j+1}\in H^{\frac{s+2-j}{5}}({\mathbb{R}}_t^{+})$, $j=0,1,2$, $s\in [0,\frac{11}{4})\setminus \{\frac{1}{2},\frac{3}{2},\frac{5}{2}\}$, 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 $g$.