Low-regularity Schr\"odinger map flow on high-dimensional periodic domains

Abstract

We study the initial-value problem for the Schr\"odinger map flow from flat torus $\mathbb{T}^d$ into compact K\"ahler manifold $\mathcal{N}$. When $d \geq 3$ and $\mathcal{N} = \mathbb{S}^2$, we establish local well-posedness in $H^{\sigma}_x$ with $\sigma > d/2 + 1/2$. In this case, the evolution equation for the gradient of the solution reduces to a certain semilinear nonlinear Schr\"odinger equation (also known as modified Schr\"odinger map flow) when formulated in orthonormal frames. For general compact K\"ahler targets, we only obtain local well-posedness in $H^{\sigma}_x$ with $ \sigma > d/2 + 5/6$ due to the quasilinear nature of the flow, but in all dimensions $d \geq 2$. To the best of our knowledge, this is the first low-regularity local well-posedness result for Schr\"odinger map flow in the periodic setting, which yields a gain of $1/2$ derivatives for $\mathbb{S}^2$ targets and $1/6$ derivatives for general K\"ahler targets compared to the classical results \cite{DW,M}. The key ingredients of our method are an $L_{t, x}^2$ bilinear estimate for the first case and an \emph{a priori} $L_t^6L_x^{\infty}$ estimate for the second case, which are both achieved by combining the mass/energy and momentum balance laws of the equation with a new type of div-curl lemma introduced by the second author.