The authors consider 2-dimensional spin models including the Ishimori system, described by the equations
for the spin and a scalar potential , with , , and the following notations for any pair of vectors of and , with .
Denoting by the 2-sphere and by , the 2-dimensional hyperbolic space, they define for the space: and the metric . Then is a metric space.
Defining the operators , and by their Fourier multipliers , and , one considers the Cauchy problem for a system equivalent to (ISH1) (ISH2)
The main results are as follows:
Large data local regularity: For large enough and , (IS1)–(IS4) has a unique solution on . Moreover the solution is maximal in the following sense: if , and if is bounded, then .
Global existence: There is a such that if and , (IS1)–(IS4) has a unique global solution , and