论文标题
在外壳类型限制下抛物面的不平等及其在周期性Zakharov系统中的不平等不平等
Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system
论文作者
论文摘要
在本文中，我们在$ \ mathbb {t}^d $，$ d \ ge3 $上建立了Zakharov System的本地适合性，在低规律性设置中。我们的结果改善了基希莫托的工作。此外，只要$ d = 3 $和$ d \ ge5 $，只要一个人使用迭代参数，结果就会达到最高$ \ varepsilon $ - 规律性。我们介绍了傅立叶限制理论的最新发展思想。我们适应性结果证明的关键要素是涉及抛物面和锥体的新三线离散傅立叶限制估计。我们通过改善波尔加因（在输入时空函数$ f $满足$ {\ rm supps} \，\ hat {f} {f} {f} \ subset \ subset \ subset \ aft {（n）{（n）{n rm v}，{（ \ frac1n \ le |ξ| \ le 1 + \ frac1n，\; |τ-|ξ|^2 | \ le \ frac1 {n^{2}} \} $对于大$ n \ ge1 $。
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 \{ (ξ,τ) \in \mathbb{R}^{d+1}: 1- \frac1N \le |ξ| \le 1 + \frac1N,\; |τ- |ξ|^2| \le \frac1{N^{2}} \} $ for large $N\ge1$.
