This talk is based on a joint work with Shinya Kinoshita (Saitama) and Akansha Sanwal (Bielefeld). Our purpose is to establish the local well-posedness of the periodic Zakharov system with low regularity on torus $\mathbb{T}^d$ with $d\geq 3$. The sharp regularity ensuring the local well-posedness is known for $d=1$ due to Takaoka and $d=2$ due to Kishimoto while the problem for $d\ge3$ is open despite of the progress by Kishimoto. We attack to this problem by introducing recent developments of so-called decoupling inequality (Wolff’s inequality) from Fourier restriction theory and improve Kishimoto’s local well-posedness result for $d\geq 3$. At the same time, the study of the periodic Zakharov system raises a question on some trilinear estimates involving free solutions of Schrödinger and wave equations on torus which can be regarded as some trilinear decoupling type estimates involving paraboloid and cone.

Note: This seminar will be held as a Zoom online seminar.