Local well-posedness for dispersive equations with bounded data

2024/09/07 by Zhao, Jason
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2409.04706

openalex publication_date 2024/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given sufficiently regular data without decay assumptions at infinity, we prove local well-posedness for non-linear dispersive equations of the form ∂_t u + \mathsf A(∇) u + \mathcal Q(|u|²) ⋅ ∇ u= \mathcal N (u, u), where \mathsf A(∇) is a Fourier multiplier with purely imaginary symbol of order σ+ 1 for σ> 0, and polynomial-type non-linearities \mathcal Q(|u|²) and \mathcal N(u, u). Our approach revisits the classical energy method by applying it within a class of local Sobolev-type spaces ℓ^∞_{\mathsf A(ξ)} H^s (\mathbb R^d) which are adapted to the dispersion relation in the sense that functions u localised to dyadic frequency |ξ| ≈ N have size ||u||_ℓ^∞_{\mathsf A(ξ)} H^s ≈ N^s sup_diam(Q) = N^σ ||u||_{L²x (Q)}. In analogy with the classical H^s-theory, we prove ℓ^∞_{\mathsf A(ξ)} H^s-local well-posedness for s > \tfracd2 + 1 for the derivative non-linear equation, and s > \tfracd2 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.

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