On well-posedness of some one-dimensional dispersive equations with analytic nonlinearity (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

This note is a résumé of the result obtained in [22]. We consider the Cauchy problem for one-dimensional dispersive equations with a general nonlinearity. As a typical model, our equation includes the generalized Benjamin-Ono equation with the generalized dispersion: ∂ₜ𝑢 − ∂ᵪ[α]𝐷ᵪ𝑢 + ∂ᵪ(𝑢ᴷ⁺¹) = 0 for α ∈ [1, 2] and 1 ≤ 𝑘 ∈ ℕ. We obtain the unconditional local well-posedness of the Cauchy problem in 𝐻ˢ(𝕋) for 𝑠 ≥ 1 − α/2(α+1) in the periodic case. We also discuss the non-periodic problem, which is omitted in [22].