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

TL;DR

This paper establishes local well-posedness and regularity properties for the initial-boundary value problem of the fifth order Korteweg-de Vries equation, detailing precise conditions on initial and boundary data for solutions to exist and be unique.

Contribution

It provides the first rigorous proof of local well-posedness for the IBVP of the fifth order KdV with specific boundary conditions and regularity assumptions, including compatibility conditions.

Findings

Solutions exist and are unique under specified regularity and compatibility conditions.

The nonlinear component of the solution exhibits higher regularity than the initial data.

The results cover a range of Sobolev space regularities, excluding certain critical indices.

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 $g$.