We consider the initial value problem for the Landau–Lifshitz equation with a helicity term (chiral interaction term), which arises from the Dzyaloshinskii–Moriya interaction energy. We prove that it is well-posed locally in-time in the space [Formula: see text] for s ≥ 3 with [Formula: see text] and [Formula: see text]. We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space [Formula: see text] for s &gt; 2 with [Formula: see text]. Our proof is based on two different approaches: One is the geometric energy method by McGahagan and the other is via the modified Schrödinger map equation. In the present analysis, we exploit a special structure of the helicity term, which enables us to overcome the difficulty in the quadratic derivative nonlinearity of the helicity term. 

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Local well-posedness of the Landau–Lifshitz equation with helicity term

We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We prove that it is well-posed locally-in-time in the space $\bar{k} +H^s$ for $s\ge 3$ with $s\in \mathbb{Z}$ and $\bar{k}={}^t(0,0,1)$. We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space $\bar{k} + H^s$ for $s>2$ with $s\in \mathbb{R}$. Our proof is base on the analysis via the modified Schrodinger map equation.

Local well-posedness for the Landau-Lifshitz equation with helicity term

We consider the initial-value problem for the equivariant Schrodinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537-583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbf{R}^2)\cap \dot{H}^2(\mathbf{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schrodinger map equation in low regularity settings without smallness of energy.

Ikkei Shimizu

Papers

Remarks on local theory for Schrödinger maps near harmonic maps