Low-regularity Schrödinger maps. (English) Zbl 1212.35449

Ionescu, Alexandru D.; Kenig, Carlos E.

Differ. Integral Equ. 19, No. 11, 1271-1300 (2006).

Summary: We prove that the Schrödinger map initial-value problem \(\partial _{t}s=s\times \Delta _{x}s\) on \(\mathbb R^d\times [-1,1]\), \(s(0)=s_0\) is locally well posed for small data \(s_0\in H_{Q}^{\sigma _0}(\mathbb R^d;\mathbb S^2)\), \(\sigma _0>(d+1)/2\), \(Q\in \mathbb S^2\).

MSC:

35Q55

NLS equations (nonlinear Schrödinger equations)

Keywords:

Schrödinger map; initial-value problem; local well posedness