Improved bilinear Strichartz estimates with application to the well-posedness of periodic generalized KdV type equations

Abstract

We improve our previous result ([29]) on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same hypotheses that the dispersive operator behaves for high frequencies as a Fourier multiplier by i|ξ| α ξ, with 1 ≤ α ≤ 2, and that the nonlinear term is of the form ∂ x f (u) where f is the sum of an entire series with infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in H s (T) for s ≥ 1 − α 4 with s > 1/2. It is worth noticing that this result is optimal in the case α = 2 (generalized KdV equation) in view of the restriction s > 1/2 for the continuous injection of H s (T) into L ∞ (T). Our main new ingredient is the introduction of improved bilinear estimates in the spirit of the improved Strichartz estimates introduced by Koch-Tzvetkov. Finally note that this enables us to derive global existence results for α ∈ [4/3, 2].