Local Well-Posedness for the Zakharov System in Dimension $d\leqslant 3$

TL;DR

This paper proves local well-posedness of the Zakharov system in low dimensions by developing new solution spaces with temporal weights, extending previous results and achieving sharpness up to endpoints.

Contribution

It introduces modified $X^{s,b}$ spaces with temporal weights to establish local well-posedness for the Zakharov system in dimensions up to three, improving prior results.

Findings

Established local well-posedness in Sobolev spaces for $d \,\leqslant\, 3$

Developed new solution spaces with temporal weights

Result is sharp up to endpoints

Abstract

The Zakharov system in dimension $d\leqslant 3$ is shown to be locally well-posed in Sobolev spaces $H^s \times H^l$, extending the previously known result. We construct new solution spaces by modifying the $X^{s,b}$ spaces, specifically by introducing temporal weights. We use contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.