On the well-posedness of a quasi-linear Korteweg-de Vries equation

TL;DR

This paper establishes local well-posedness for a quasi-linear generalization of the KdV equation, utilizing energy estimates and structural properties to prove existence, uniqueness, and continuous dependence on initial data.

Contribution

It provides the first well-posedness result for a quasi-linear KdV equation with a general Hamiltonian structure, employing novel energy estimate techniques and gauge methods.

Findings

Proves local existence and uniqueness of solutions.

Demonstrates continuous dependence on initial data.

Uses structural properties to control subprincipal terms.

Abstract

The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamil- tonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major inter- est being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim & Ponce (SIAM J. Math. Anal., 2002) and developed later by Kenig, Ponce & Vega (Invent. Math., 2004) and S. Benzoni-Gavage, R. Danchin & S. Descombes (Electron. J. Diff. Eq., 2006). The existence of a solution is obtained as a limit from regularized parabolic problems. Uniqueness and continuity with respect to the initial data are proven using a Bona-Smith regularization technique.