On the Propagation of Regularity of Solutions to the KdV Equation on the positive Half-line

TL;DR

This paper investigates how regularity properties of solutions to the KdV equation on the positive half-line propagate over time, revealing infinite speed regularity transfer and boundary effects.

Contribution

It establishes the propagation of regularity for KdV solutions on the half-line and introduces a stopping time due to boundary influences.

Findings

Regularity propagates with infinite speed to the left over time.

Existence of a finite stopping time T* influenced by boundary data.

Gain in regularity of trace derivatives of solutions.

Abstract

We study special regularity properties of solutions to the initial-boundary value problem associated with the Korteweg-de Vries equations posed on the positive half-line. In particular, for initial data $u_0 \in H^{\frac{3}{4}^{+}}(\mathbb{R}^+)$ and boundary data $f\in H^{\frac32^+}(\R^+)$, where the restriction of $u_0$ to some subset of $(b,\infty)$ has an extra regularity for any $b>0$, we prove that the regularity of solutions $u$ moves with infinite speed to its left as time evolves until a certain time $T^*$. The existence of a stopping time $T^{*}$ appears because of the effect of the boundary function $f$. Also, as a consequence of our proof, we prove a gain in the regularity of the trace derivatives of the solutions for the Korteweg-de Vries on the half-line.