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Zbl 1215.35145
Bejenaru, I.; Ionescu, A.D.; Kenig, C.E.
On the stability of certain spin models in $2+1$ dimensions. (English)
[J] J. Geom. Anal. 21, No. 1, 1-39 (2011). ISSN 1050-6926; ISSN 1559-002X/e

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 $\vec s(x,y,t)\in\Bbb S^2$ and a scalar potential $\zeta(x,y,t)\in\Bbb R$, with $\varepsilon=\pm 1$, $\mu=\pm 1$, and the following notations for any pair of vectors of $\Bbb R^3: \vec u\cdot_{\mu}\vec v= {^t}\vec u\,\eta_\mu\,\vec v$ and $\vec u\times_\mu\vec v=\eta_\mu(\vec u \times \vec v)$, with $\eta_\mu:=\text {diag}(\mu,1,1)$. Denoting by $S_1:=\Bbb S^2=\{(x,y,z)\in\Bbb R^3: x^2+y^2+z^2=1\}$ the 2-sphere and by $S_{-1}:=\Bbb H^2=\{(x,y,z)\in\Bbb R^3: x^2-y^2-z^2=1$, $x>0\}$ 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) $$\align \partial_t \vec s&= \vec s\times_\mu (\partial_x^2 \vec s+\varepsilon\partial_y^2 \vec s) +\partial_x\vec s\ \partial_y \zeta-\varepsilon\partial_y \vec s\ \partial_x \zeta, \tag{IS1}\\ \partial_x\zeta&= -R_x \nabla^{-1}\big[2\mu\vec s \cdot_\mu(\partial_x \vec s \times_\mu \partial_y \vec s)\big], \tag{IS2}\\ \partial_y\zeta&= -R_y \nabla^{-1}\big[2\mu\vec s \cdot_\mu (\partial_x \vec s \times_\mu \partial_y \vec s)\big], \tag{IS3}\\ \vec s(x,y,0)&= \vec f. \tag{IS4} \endalign$$ 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 $I(\vec f)\subset\Bbb R$. 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 $|||D \vec s|||_{L^4_{x,y,t}(\Bbb R^2\times I_+(\vec f))}=\infty$. 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 $\vec s\in C(\Bbb R;\widetilde H^\sigma_\mu)$, and $$|||D \vec s|||_{L_t^\infty L^2_{x,y}(\Bbb R^2\times\Bbb R)} +|||D \vec s|||_{L^4_{x,y,t} (\Bbb R^2\times\Bbb R)} \leq |||D \vec s|||_{L^2}.$$
[Bernard Ducomet (Bruyères le Châtel)]

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
81Q05 Closed and approximate solutions to quantum-mechanical equations

Keywords: spin models; Ishimori system; orthonormal frame; focusing; defocusing

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