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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

t s =s × μ ( x 2 s +ε y 2 s )+ x s y ζ-ε y s x ζ,( ISH 1)
x 2 ζ+ y 2 ζ=2s · μ x s × μ y s ,( ISH 2)

for the spin s (x,y,t)𝕊 2 and a scalar potential ζ(x,y,t), with ε=±1, μ=±1, and the following notations for any pair of vectors of 3 :u · μ v = t u η μ v and u × μ v =η μ (u ×v ), with η μ :=diag(μ,1,1).

Denoting by S 1 :=𝕊 2 ={(x,y,z) 3 :x 2 +y 2 +z 2 =1} the 2-sphere and by S -1 := 2 ={(x,y,z) 3 :x 2 -y 2 -z 2 =1, x>0} the 2-dimensional hyperbolic space, they define for σ1 the space: H ˜ μ σ :={f C b 1 ( 2 :S μ ): x f , y f H σ-1 }, and the metric d σ (f,g)=f-g + x (f-g) H σ-1 + y (f-g) H σ-1 . Then (H ˜ μ σ ,d σ ) is a metric space.

Defining the operators -1 , R x and R y by their Fourier multipliers i1 |ξ|, iξ x |ξ| and iξ y |ξ|, one considers the Cauchy problem for a system equivalent to (ISH1) (ISH2)

t s =s × μ ( x 2 s +ε y 2 s )+ x s y ζ-ε y s x ζ, x ζ=-R x -1 2 μ s · μ ( x s × μ y s ), y ζ=-R y -1 2 μ s · μ ( x s × μ y s ),s (x,y,0)=f ·

The main results are as follows:

Large data local regularity: For σ large enough and f H ˜ μ σ , (IS1)–(IS4) has a unique solution on I(f ). Moreover the solution is maximal in the following sense: if |Ds |:=[ x s · μ x s + y s · μ y s ] 1/2 , and if I + (f ):=I(f )[0,) is bounded, then |||Ds ||| L x,y,t 4 ( 2 ×I + (f )) =.

Global existence: There is a δ 0 >0 such that if f H ˜ μ σ and |||Ds ||| L 2 δ 0 , (IS1)–(IS4) has a unique global solution s C(;H ˜ μ σ ), and

|||Ds ||| L t L x,y 2 ( 2 ×) +|||Ds ||| L x,y,t 4 ( 2 ×) |||Ds ||| L 2 ·

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
81Q05Closed and approximate solutions to quantum-mechanical equations
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