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A priori bounds for the 1D cubic NLS in negative Sobolev spaces. (English) Zbl 1169.35055
The authors consider the cubic nonlinear Schrödinger equation in one space dimension and conjecture that it is locally well-posed for initial data in \(H^s\) with \(s\geq -\frac{1}{6}\). To obtain such a result one needs to establish a priori \(H^s\) estimates for the solutions and then prove continuous dependence on the initial data. In this paper the authors solve the first half of the problem: They show that the solutions satisfy a priori local in time \(H^s\) estimates in terms of the \(H^s\) norm of the initial data for \(s\geq -\frac{1}{6}\). The proof is essentially made up of three pieces: an appropiate estimate for the solutions of the nonhomogeneous linear equation, another one (fitting the first) for the nonlinear term, and a propagation principle for the norm of the solutions of the nonlinear equation. These are assembled in a continuity argument.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
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