Low-regularity Schrödinger map flow on high-dimensional periodic domains

Abstract

We study the initial-value problem for the Schrödinger map flow from flat torus ${\mathbb{T}}^d$ into compact Kähler manifold $\mathcal{N}$. When $d\ge 3$ and $\mathcal{N}={\mathbb{S}}^2$, we establish local well-posedness in $H^{\ddot{I}}{ƒ}_x$ with $\ddot{I}ƒ>d/2+1/2$. In this case, the evolution equation for the gradient of the solution reduces to a certain semilinear nonlinear Schrödinger equation (also known as modified Schrödinger map flow) when formulated in orthonormal frames. For general compact Kähler targets, we only obtain local well-posedness in $H^{\ddot{I}}{ƒ}_x$ with $\ddot{I}ƒ>d/2+5/6$ due to the quasilinear nature of the flow, but in all dimensions $d\ge 2$. To the best of our knowledge, this is the first low-regularity local well-posedness result for Schrödinger 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 Kähler 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} $L_t^6{L}_x^{\infty }$ 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.

48 pages, all comments are welcome