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On the stability of certain spin models in $2+1$ dimensions. (English) Zbl 1215.35145

The authors consider 2-dimensional spin models including the Ishimori system, described by the equations

${\partial }_{t}\stackrel{\to }{s}=\stackrel{\to }{s}{×}_{\mu }\left({\partial }_{x}^{2}\stackrel{\to }{s}+\epsilon {\partial }_{y}^{2}\stackrel{\to }{s}\right)+{\partial }_{x}\stackrel{\to }{s}\phantom{\rule{4pt}{0ex}}{\partial }_{y}\zeta -\epsilon {\partial }_{y}\stackrel{\to }{s}\phantom{\rule{4pt}{0ex}}{\partial }_{x}\zeta ,\phantom{\rule{2.em}{0ex}}\left(\mathrm{ISH}1\right)$
${\partial }_{x}^{2}\zeta +{\partial }_{y}^{2}\zeta =2\stackrel{\to }{s}{·}_{\mu }\left({\partial }_{x}\stackrel{\to }{s}{×}_{\mu }{\partial }_{y}\stackrel{\to }{s}\right),\phantom{\rule{2.em}{0ex}}\left(\mathrm{ISH}2\right)$

for the spin $\stackrel{\to }{s}\left(x,y,t\right)\in {𝕊}^{2}$ and a scalar potential $\zeta \left(x,y,t\right)\in ℝ$, with $\epsilon =±1$, $\mu =±1$, and the following notations for any pair of vectors of ${ℝ}^{3}:\stackrel{\to }{u}{·}_{\mu }\stackrel{\to }{v}={}^{t}\stackrel{\to }{u}\phantom{\rule{0.166667em}{0ex}}{\eta }_{\mu }\phantom{\rule{0.166667em}{0ex}}\stackrel{\to }{v}$ and $\stackrel{\to }{u}{×}_{\mu }\stackrel{\to }{v}={\eta }_{\mu }\left(\stackrel{\to }{u}×\stackrel{\to }{v}\right)$, with ${\eta }_{\mu }:=\text{diag}\left(\mu ,1,1\right)$.

Denoting by ${S}_{1}:={𝕊}^{2}=\left\{\left(x,y,z\right)\in {ℝ}^{3}:{x}^{2}+{y}^{2}+{z}^{2}=1\right\}$ the 2-sphere and by ${S}_{-1}:={ℍ}^{2}=\left\{\left(x,y,z\right)\in {ℝ}^{3}:{x}^{2}-{y}^{2}-{z}^{2}=1$, $x>0\right\}$ the 2-dimensional hyperbolic space, they define for $\sigma \ge 1$ the space: ${\stackrel{˜}{H}}_{\mu }^{\sigma }:=\left\{\stackrel{\to }{f}\in {C}_{b}^{1}\left({ℝ}^{2}:{S}_{\mu }\right):\phantom{\rule{4pt}{0ex}}{\partial }_{x}\stackrel{\to }{f},{\partial }_{y}\stackrel{\to }{f}\in {H}^{\sigma -1}\right\},$ and the metric ${d}_{\sigma }\left(f,g\right)={\parallel f-g\parallel }_{\infty }+\parallel {\partial }_{x}{\left(f-g\right)\parallel }_{{H}^{\sigma -1}}+{\parallel {\partial }_{y}\left(f-g\right)\parallel }_{{H}^{\sigma -1}}$. Then $\left({\stackrel{˜}{H}}_{\mu }^{\sigma },{d}_{\sigma }\right)$ 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)

$\begin{array}{cc}\hfill {\partial }_{t}\stackrel{\to }{s}& =\stackrel{\to }{s}{×}_{\mu }\left({\partial }_{x}^{2}\stackrel{\to }{s}+\epsilon {\partial }_{y}^{2}\stackrel{\to }{s}\right)+{\partial }_{x}\stackrel{\to }{s}\phantom{\rule{4pt}{0ex}}{\partial }_{y}\zeta -\epsilon {\partial }_{y}\stackrel{\to }{s}\phantom{\rule{4pt}{0ex}}{\partial }_{x}\zeta ,\hfill \\ \hfill {\partial }_{x}\zeta & =-{R}_{x}{\nabla }^{-1}\left[2\mu \stackrel{\to }{s}{·}_{\mu }\left({\partial }_{x}\stackrel{\to }{s}{×}_{\mu }{\partial }_{y}\stackrel{\to }{s}\right)\right],\hfill \\ \hfill {\partial }_{y}\zeta & =-{R}_{y}{\nabla }^{-1}\left[2\mu \stackrel{\to }{s}{·}_{\mu }\left({\partial }_{x}\stackrel{\to }{s}{×}_{\mu }{\partial }_{y}\stackrel{\to }{s}\right)\right],\hfill \\ \hfill \stackrel{\to }{s}\left(x,y,0\right)& =\stackrel{\to }{f}·\hfill \end{array}$

The main results are as follows:

Large data local regularity: For $\sigma$ large enough and $\stackrel{\to }{f}\in {\stackrel{˜}{H}}_{\mu }^{\sigma }$, (IS1)–(IS4) has a unique solution on $I\left(\stackrel{\to }{f}\right)\subset ℝ$. Moreover the solution is maximal in the following sense: if $|D\stackrel{\to }{s}|:={\left[{\partial }_{x}\stackrel{\to }{s}{·}_{\mu }{\partial }_{x}\stackrel{\to }{s}+{\partial }_{y}\stackrel{\to }{s}{·}_{\mu }{\partial }_{y}\stackrel{\to }{s}\right]}^{1/2}$, and if ${I}_{+}\left(\stackrel{\to }{f}\right):=I\left(\stackrel{\to }{f}\right)\cap \left[0,\infty \right)$ is bounded, then $|||D\stackrel{\to }{s}{|||}_{{L}_{x,y,t}^{4}\left({ℝ}^{2}×{I}_{+}\left(\stackrel{\to }{f}\right)\right)}=\infty$.

Global existence: There is a ${\delta }_{0}>0$ such that if $\stackrel{\to }{f}\in {\stackrel{˜}{H}}_{\mu }^{\sigma }$ and $|||D\stackrel{\to }{s}{|||}_{{L}^{2}}\le {\delta }_{0}$, (IS1)–(IS4) has a unique global solution $\stackrel{\to }{s}\in C\left(ℝ;{\stackrel{˜}{H}}_{\mu }^{\sigma }\right)$, and

$|||D\stackrel{\to }{s}{|||}_{{L}_{t}^{\infty }{L}_{x,y}^{2}\left({ℝ}^{2}×ℝ\right)}+|||D\stackrel{\to }{s}{|||}_{{L}_{x,y,t}^{4}\left({ℝ}^{2}×ℝ\right)}\le |||D\stackrel{\to }{s}{|||}_{{L}^{2}}·$

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