# On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation

## TL;DR

This paper proves that solutions to the fractional Korteweg-de Vries equation propagate initial regularity from right to left, extending known results from classical KdV and Benjamin-Ono equations to fractional cases.

## Contribution

It introduces a method to demonstrate propagation of regularity for the fractional KdV, generalizing previous results to equations with fractional dispersion.

## Key findings

- Regularity propagates infinitely fast to the left over time.
- The approach adapts commutator decomposition techniques from previous dispersive equations.
- Solutions exhibit propagation of regularity similar to classical dispersive equations.

## Abstract

We consider the initial value problem (IVP) for the fractional Korteweg-de Vries equation (fKdV) \begin{equation}\label{abstracteq1} \left\{ \begin{array}{ll} \partial_{t}u-D_{x}^{\alpha}\partial_{x}u+u\partial_{x}u=0, & x,t\in\mathbb{R},\,0<\alpha<1, \\ u(x,0)=u_{0}(x).& \\ \end{array} \right. \end{equation} It has been shown that the solutions to certain dispersive equations satisfy the propagation of regularity phenomena. More precisely, it deals in determine whether regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. This property was found originally in solutions of Korteweg-de Vries (KdV) equation and it has been also verified in other dispersive equations as the Benjamin-Ono (BO) equation. Recently, it has been shown that the solutions of the dispersive generalized Benjamin-Ono (DGBO) equation, this is $\alpha\in (2,3)$ in \eqref{abstracteq1}; also satisfy the propagation of regularity phenomena. This is achieved by introducing a commutator decomposition to handle the dispersive part in the equation. Following the approach used in the DGBO, we prove that the solutions of the fKdV also satisfies the propagation of regularity phenomena. Consequently, this type of regularity travels with infinite speed to its left as time evolves.