On the Cauchy problem for dispersive Burgers type equations

We study the paralinearised weakly dispersive Burgers type equation: ∂tu+ ∂x[Tuu]− T ∂xu 2 u+ ∂x |D| α−1 u = 0, α ∈]1, 2[, which contains the main non linear ”worst interaction” terms, i.e low-high interaction terms, of the usual weakly dispersive Burgers type equation: ∂tu+ u∂xu+ ∂x |D| α−1 u = 0, α ∈]1, 2[, with u0 ∈ H (D), where D = T or R. Through a paradifferential complex Cole-Hopf type gauge transform we introduced in [38], we prove a new a priori estimate in H(D) under the control of ∥