Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations

Abstract

We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity in the periodic setting. Our main hypotheses are both that the dispersive operator behaves for high frequencies as a Fourier multiplier by i|ξ|αξ, with 1≤α≤2, and that the nonlinear term is of the form ∂xf(u) where f is the sum of an entire series with infinite radius of convergence. Under these conditions, we prove the unconditional local well-posedness of the Cauchy problem in Hs(T) for s≥1−α2(α+1). This leads to some global existence results in the energy space Hα/2(T), for α∈[2,2].