Large data local well-posedness for a class of KdV-type equations II

TL;DR

This paper establishes local well-posedness for a class of KdV-type equations with polynomial nonlinearities in low regularity Sobolev spaces, extending previous results to include certain quadratic terms.

Contribution

It introduces new analytical techniques to prove well-posedness for KdV-type equations with quadratic nonlinearities in low regularity spaces.

Findings

Proved local well-posedness in low regularity Sobolev spaces.

Extended well-posedness results to certain quadratic nonlinearities.

Utilized spaces and estimates similar to those for quasilinear Schrödinger equations.

Abstract

We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic terms, Kenig-Ponce-Vega proved local well-posedness in H^s for large s. In this paper we prove local well-posedness in low regularity Sobolev spaces and extend the result to certain quadratic nonlinearities. The result is based on spaces and estimates similar to those used by Marzuola-Metcalfe-Tataru for quasilinear Schrodinger equations.