Local well-posedness for the Zakharov system in dimension <i>d</i> ≤ 3

Abstract

<p style='text-indent:20px;'>The Zakharov system in dimension <inline-formula><tex-math id="M1">\begin{document}$ d\leqslant 3 $\end{document}</tex-math></inline-formula> is shown to be locally well-posed in Sobolev spaces <inline-formula><tex-math id="M2">\begin{document}$ H^s \times H^l $\end{document}</tex-math></inline-formula>, extending the previously known result. We construct new solution spaces by modifying the <inline-formula><tex-math id="M3">\begin{document}$ X^{s,b} $\end{document}</tex-math></inline-formula> spaces, specifically by introducing temporal weights. We use the contraction mapping principle to prove local well-posedness in the same. The result obtained is sharp up to endpoints.</p>