Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$
Year: 2025
Type: preprint
Abstract: We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$, \begin{equation} \|e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} ϕ\|_{L^4_{t,x_{1},x_{2}}([0,1]\times \mathbb{R} \times \mathbb{T})}\lesssim \|ϕ\|_{L_{x_{1},x_{2}}^2(\mathbb{R} \times \mathbb{T})}, \end{equation} which serves as the hyperbolic analogue of the classical result of Takaoka-Tzvetkov \cite{takaoka20012d}. The proof is based on the combination of a robust kernel decomposition method with precise measure estimates for semi-algebraic sets. As an immediate application, we establish the global well-posedness for the cubic hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data.
Source: arXiv (Cornell University)
Authors: Deng, Yangkendi, Fan, Chenjie, Zhao, Zehua
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7106216342
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