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

We improve our previous result ([L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022) 109490]) 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 [Formula: see text] with [Formula: see text], and that the nonlinear term is of the form [Formula: see text] where [Formula: see text] is a real analytic function whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in [Formula: see text] for [Formula: see text] with [Formula: see text]. It is worth noting that this result is optimal in the case [Formula: see text] (generalized KdV equation) in view of the restriction [Formula: see text] for the continuous injection of [Formula: see text] into [Formula: see text]. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani [A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Anal. PDE 5 (2012) 339–363] in the context of Schrödinger equations on a compact manifold. Finally, the main theorem yields global existence results for [Formula: see text].