Local well-posedness for dispersive equations with bounded data

Abstract

Given sufficiently regular data \textit{without} decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form \[ \partial_t u + \mathsf A(\nabla) u + \mathcal Q(|u|^2) \cdot \nabla u= \mathcal N (u, \overline u), \] where $\mathsf{A}(\nabla )$ is a Fourier multiplier with purely imaginary symbol of order $\ddot{I}ƒ+1$ for $\ddot{I}ƒ>0$, and polynomial-type non-linearities $\mathcal{Q}(|u{|}^2)$ and $\mathcal{N}(u,\overline{u})$. Our approach revisits the classical energy method by applying it within a class of local Sobolev-type spaces ${\ell }_{\mathsf{A}(\hat{I}¾)}^{\infty }{H}^s({\mathbb{R}}^d)$ which are adapted to the dispersion relation in the sense that functions $u$ localised to dyadic frequency $|\hat{I}¾|\approx N$ have size \[ ||u||_{\ell^\infty_{\mathsf A(ξ)} H^s} \approx N^s \sup_{{\operatorname{diam}(Q) = N^σ}} ||u||_{L^2_x (Q)}. \] In analogy with the classical $H^s$-theory, we prove ${\ell }_{\mathsf{A}(\hat{I}¾)}^{\infty }{H}^s$-local well-posedness for $s>\frac{d}{2}+1$ for the derivative non-linear equation, and $s>\frac{d}{2}$ without the derivative non-linearity. As an application, we show that if in addition the initial data is spatially almost periodic, then the solution is also spatially almost periodic.

26 pages