Regularity Results on the Flow Maps of Periodic Dispersive Burgers Type Equations and the Gravity–Capillary Equations

In the first part of this paper we prove that flow associated to a dispersive Burgers equation with non local term form $$\left| D \right| ^{\alpha -1} \partial _x u$$ , $$\alpha \in [1,+\infty [$$ is Lipschitz from bounded sets $$H^s_0({\mathbb {T}};{\mathbb {R}})$$ $$C^0([0,T],H^{s-(2-\alpha )^+}_0({\mathbb {R}}))$$ for $$T>0$$ and $$s>\big \lceil \frac{\alpha }{\alpha -1}\big \rceil -\frac{1}{2}$$ where $$H^s_0$$ are Sobolev spaces functions 0 mean value, proving result obtained in Said (A geometric proof quasi-linearity water-waves system incompressible Euler equations) optimal on torus. The relies paradifferential generalization complex Cole–Hopf gauge transformation introduced by Tao (J Hyperbol Differ Equ 1:27–49, 2004) Benjamin–Ono equation. For Baker–Campbell–Hausdorff formula flows hyperbolic equations stability class operators modulo more regular remainders, under conjugation such flows. new characterization spirit Beals (Duke Math J 44:45–57, 1977). second use version previous method re-normalization one dimensional periodic gravity–capillary $$H^s$$ $$C^0([0,T],H^{s-\frac{1}{2}})$$ $$s>3+\frac{1}{2}$$ . This proves water waves system.