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Global existence and uniqueness of Schrödinger maps in dimensions d4. (English) Zbl 1152.35049

For σ0 and n{1,2,} let H σ =H σ ( d , n ) denote the Banach space of n -valued Sobolev functions on d . For σ0 and Q=(Q 1 ,Q 2 ,Q 3 )𝕊 2 define complete metric space H Q σ =H Q σ ( d ;𝕊 2 3 )={f: d 3 ;|f(x)|1,f-QH σ } with induced distance d Q σ (f,g)=f-g H σ , and H Q = σ + H Q σ . Let s: d ×𝕊 2 3 is a continuous function. The authors consider the Schrödinger map initial-value problem

s=s×Δ x s,on d ×s(0)=s 0 ·

It is proved that in dimensions d4 this problem admits a unique global (in time) solution sC(:H Q ), provided that s 0 H Q and s 0 -Q H d/2 1, where Q𝕊 2 .


MSC:
35K55Nonlinear parabolic equations
35K15Second order parabolic equations, initial value problems
35K60Nonlinear initial value problems for linear parabolic equations
46E35Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H99Nonlinear operators