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A refined global well-posedness result for Schrödinger equations with derivative. (English) Zbl 1034.35120

Summary: We prove that the one-dimensional Schrödinger equation with derivative in the nonlinear term is globally well-posed in \(H^{s}\) for \(s >\frac 1 2\) for data small in \(L^{2}\). To understand the strength of this result one should recall that for \(s<\frac 1 2\) the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ”I-method” used by the same authors to obtain global well-posedness for \(s>\frac 2 3\). The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in \(H^{s}\) for \(s>\frac 1 2\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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