Local Well Posedness of Quasi-Linear Systems Generalizing KdV

TL;DR

This paper establishes local well-posedness for a class of quasilinear dispersive PDEs extending the KdV equation, using adapted smoothing estimates and viscosity methods.

Contribution

It extends techniques from quasi-linear Schrödinger equations to third-order dispersive systems, providing new well-posedness results for generalized KdV models.

Findings

Proves local well-posedness for generalized KdV systems.

Develops a local smoothing estimate for linear dispersive problems.

Applies the artificial viscosity method successfully.

Abstract

In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig- Ponce-Vega from the Quasi-Linear Schr\"odinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.