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Abstract
In this paper, we first introduce a new function space MH^{θ,p} whose norm is given by the ℓ^psum of modulated H^θnorms of a given function. In particular, when θ<−1/2, we show that the space MH^{θ,p} agrees with the modulation space M^{2,p}(ℝ) on the real line and the FourierLebesgue space FL^p(핋) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by KillipVişanZhang to the modulation space setting. By applying the scaling symmetry, we then prove global wellposedness of the onedimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on ℝ is globally wellposed in M^{2,p}(ℝ) for any p<∞, while the renormalized cubic NLS on 핋 is globally wellposed in FL^p(핋) for any p<∞.
In Appendix, we also establish analogous globalintime bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the FourierLebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdVNLS equation.
Original language  English 

Pages (fromto)  612640 
Number of pages  29 
Journal  Journal of Differential Equations 
Volume  269 
Issue number  1 
Early online date  16 Jan 2020 
DOIs  
Publication status  Published  15 Jun 2020 
Keywords
 nonlinear Schrödinger equation
 modified KdV equation
 global wellposedness
 complete integrability
 modulation space
 FourierLebesgue space
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 1 Finished

ProbDynDispEq  Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
Project: Research