Abstract
We consider the Cauchy problem for the nonlocal (derivative) NLS in super-critical function spaces Eσs for which the norms are defined by‖f‖Eσs=‖〈ξ〉σ2s|ξ|fˆ(ξ)‖L2,s<0,σ∈R. Any Sobolev space Hr is a subspace of Eσs, i.e., Hr⊂Eσs for any r,σ∈R and s<0. Let s<0 and σ>−1/2 (σ>0) for the nonlocal NLS (for the nonlocal derivative NLS). We show the global existence and uniqueness of the solutions if the initial data belong to Eσs and their Fourier transforms are supported in (0,∞), the smallness conditions on the initial data in Eσs are not required for the global solutions.
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