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

Mietka, Colin

Annales mathématiques Blaise Pascal, Tome 24 (2017) no. 1, pp. 83-114.

The Korteweg-de Vries equation (KdV) and various generalized, most often semilinear 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 Hamiltonian 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 interest 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 [22] and developed later by Kenig, Ponce & Vega [20] and S. Benzoni-Gavage, R. Danchin & S. Descombes [4]. 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.

Classification : 35Q53
Keywords: quasilinear dispersive equation, energy estimates, gauging technique, parabolic regularization, Bona–Smith technique
Mots clés : quasilinear dispersive equation, energy estimates, gauging technique, parabolic regularization, Bona–Smith technique

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