On the Propagation of Regularity of Solutions to the KdV Equation on the positive Half-line
Márcio Cavalcante, Aílton C. Nascimento
TL;DR
This work addresses the propagation of regularity for the KdV IBVP on the half-line, showing that regularity localized in the initial data and boundary data can propagate leftward at infinite speed up to a boundary-induced stopping time. By combining weighted energy estimates, Kato smoothing, and trace estimates, the authors prove propagation results for derivatives up to order $l$, with explicit time bounds $T^*$ that depend on the boundary data and tail location. They also demonstrate a gain in the regularity of trace derivatives at the boundary, notably establishing improved regularity for $ abla_x^2 u(0,t)$ when $l=1$. The analysis extends the propagation-of-regularity phenomenon to half-line dispersive problems, highlighting the role of the boundary in delaying or truncating propagation and linking smoothing to boundary-trace behavior in nonlinear dispersive dynamics.
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.