In this talk, we consider the low regularity well-posedness problem for the Korteweg-de Vries equation (KdV) on the real line. Aiming to bridge the regularity gap between the scaling critical space and the known optimal well-posedness in $L^2$-based Sobolev spaces, we consider rough data in Fourier-Lebesgue spaces. Via infinite normal form reductions and exploiting algebraic cancellations, we introduce a new gauged KdV equation, equivalent to the original one at high regularity, but better behaved for rough solutions below the $H^{-1}$-scale. Surprisingly, our method does not rely on the completely integrable structure of KdV and is easily adapted to other equations with quadratic derivative nonlinearities, such as the dispersion-generalized Benjamin-Ono equations.

This talk is based on joint work with Simão Correia (IST, U. Lisboa) and João Pedro Ramos (IMPA).