Bejenaru, I.; Ionescu, A. D.; Kenig, C. E. On the stability of certain spin models in \(2+1\) dimensions. (English) Zbl 1215.35145 J. Geom. Anal. 21, No. 1, 1-39 (2011). The authors consider 2-dimensional spin models including the Ishimori system, described by the equations \[ \partial_t \vec s= \vec s\times_\mu (\partial_x^2 \vec s+\varepsilon\partial_y^2 \vec s) +\partial_x\vec s\;\partial_y \zeta-\varepsilon\partial_y \vec s\;\partial_x \zeta, \tag{ISH1} \]\[ \partial_x^2\zeta+\partial_y^2\zeta= 2\vec s \cdot_{\mu}\big(\partial_x \vec s \times_\mu \partial_y \vec s \big), \tag{ISH2} \]for the spin \(\vec s(x,y,t)\in\mathbb S^2\) and a scalar potential \(\zeta(x,y,t)\in\mathbb R\), with \(\varepsilon=\pm 1\), \(\mu=\pm 1\), and the following notations for any pair of vectors of \(\mathbb R^3: \vec u\cdot_{\mu}\vec v= {^t}\vec u\,\eta_\mu\,\vec v\) and \(\vec u\times_\mu\vec v=\eta_\mu(\vec u \times \vec v)\), with \(\eta_\mu:=\text{diag}(\mu,1,1)\).Denoting by \(S_1:=\mathbb S^2=\{(x,y,z)\in\mathbb R^3: x^2+y^2+z^2=1\}\) the 2-sphere and by \(S_{-1}:=\mathbb H^2=\{(x,y,z)\in\mathbb R^3: x^2-y^2-z^2=1\), \(x>0\}\) the 2-dimensional hyperbolic space, they define for \(\sigma\geq 1\) the space: \(\widetilde H^{\sigma}_{\mu}:=\{ \vec f\in C_b^1({\mathbb R}^2:S_{\mu}):\;\partial_x\vec f,\partial_y\vec f\in H^{\sigma-1}\},\) and the metric \(d_{\sigma}(f,g)=\|f-g\|_{\infty}+\|\partial_x(f-g)\|_{H^{\sigma-1}}+\|\partial_y(f-g)\|_{H^{\sigma-1}}\). Then \((\widetilde H^\sigma_\mu, d_\sigma)\) is a metric space.Defining the operators \(\nabla^{-1}\), \(R_x\) and \(R_y\) by their Fourier multipliers \(i\frac{1}{|\xi|}\), \(i\frac{\xi_x}{|\xi|}\) and \(i\frac{\xi_y}{|\xi|}\), one considers the Cauchy problem for a system equivalent to (ISH1) (ISH2) \[ \begin{aligned} \partial_t \vec s&= \vec s\times_\mu (\partial_x^2 \vec s+\varepsilon\partial_y^2 \vec s) +\partial_x\vec s\;\partial_y \zeta-\varepsilon\partial_y \vec s\;\partial_x \zeta, \tag{IS1}\\ \partial_x\zeta&= -R_x \nabla^{-1}\big[2\mu\vec s \cdot_\mu(\partial_x \vec s \times_\mu \partial_y \vec s)\big], \tag{IS2}\\ \partial_y\zeta&= -R_y \nabla^{-1}\big[2\mu\vec s \cdot_\mu (\partial_x \vec s \times_\mu \partial_y \vec s)\big], \tag{IS3}\\ \vec s(x,y,0)&= \vec f. \tag{IS4} \end{aligned} \]The main results are as follows:Large data local regularity: For \(\sigma\) large enough and \(\vec f\in\widetilde H^\sigma_\mu\), (IS1)–(IS4) has a unique solution on \(I(\vec f)\subset\mathbb R\). Moreover the solution is maximal in the following sense: if \(|D \vec s|:=[\partial_x \vec s \cdot_\mu \partial_x \vec s+\partial_y \vec s \cdot_\mu \partial_y \vec s]^{1/2}\), and if \(I_+(\vec f):=I(\vec f)\cap[0,\infty)\) is bounded, then \(|||D \vec s|||_{L^4_{x,y,t}(\mathbb R^2\times I_+(\vec f))}=\infty\). Global existence: There is a \(\delta_0>0\) such that if \(\vec f\in\widetilde H^\sigma_\mu\) and \(|||D \vec s|||_{L^2}\leq \delta_0\), (IS1)–(IS4) has a unique global solution \(\vec s\in C(\mathbb R;\widetilde H^\sigma_\mu)\), and \[ |||D \vec s|||_{L_t^\infty L^2_{x,y}(\mathbb R^2\times\mathbb R)} +|||D \vec s|||_{L^4_{x,y,t} (\mathbb R^2\times\mathbb R)} \leq |||D \vec s|||_{L^2}. \] Reviewer: Bernard Ducomet (Bruyères le Châtel) Cited in 4 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics Keywords:spin models; Ishimori system; orthonormal frame; focusing; defocusing PDFBibTeX XMLCite \textit{I. Bejenaru} et al., J. Geom. Anal. 21, No. 1, 1--39 (2011; Zbl 1215.35145) Full Text: DOI arXiv References: [1] Bejenaru, I., Ionescu, A.D., Kenig, C.E.: Global existence and uniqueness of Schrödinger maps in dimensions d. Adv. 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