Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system

TL;DR

This paper advances the understanding of the Zakharov system on high-dimensional tori by establishing sharp local well-posedness results through novel Fourier restriction estimates involving paraboloids and cones.

Contribution

It introduces a new trilinear Fourier restriction estimate that improves decoupling inequalities under shell-type restrictions, leading to sharper well-posedness results for the Zakharov system.

Findings

Established local well-posedness for Zakharov system on $\

$ ext{d} ext{ } ext{ge} 3$ in low regularity settings.

Improved Bourgain--Demeter's decoupling range for paraboloid under shell restriction.

Abstract

In this paper, we establish local well-posedness for the Zakharov system on $\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves the work of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss of regularity when $d=3$ and $d\ge5$ as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain--Demeter's range of exponent for the linear decoupling inequality for paraboloid under the constraint that the input space-time function $f$ satisfies ${\rm supp}\, \hat{f} \subset \{ (\xi,\tau) \in \mathbb{R}^{d+1}: 1- \frac1N \le |\xi| \le 1 + \frac1N,\; |\tau - |\xi|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.