Local well-posedness for dispersive equations with bounded data

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 $\sigma + 1$ for $\sigma > 0$, 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 $\ell^\infty_{\mathsf A(\xi)} H^s (\mathbb R^d)$ which are adapted to the dispersion relation in the sense that functions $u$ localised to dyadic frequency $|\xi| \approx N$ have size $$||u||_{\ell^\infty_{\mathsf A(\xi)} H^s} \approx N^s \sup_{{\operatorname{diam}(Q) = N^\sigma}} ||u||_{L^2_x (Q)}.$$ In analogy with the classical $H^s$-theory, we prove $\ell^\infty_{\mathsf A(\xi)} H^s$-local well-posedness for $s > \tfrac{d}2 + 1$ for the derivative non-linear equation, and $s > \tfrac{d}2$ 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.