Scattering and Smoothing for the Two-Dimensional Cubic Nonlinear Schrödinger Equation

Séminaire d’Analyse
Scattering theory describes the large-time behavior of nonlinear evolution equations through comparison with a linear flow. This raises a further question: can the nonlinear correction to the asymptotic state be smoother than the initial data?I will begin with an introduction to scattering for nonlinear Schrödinger equations, highlighting several developments that motivate this question. I will then present a deterministic smoothing result for the two-dimensional cubic equation. Assuming that a solution scatters in the harmonic Sobolev space $(mathcal H^s)$, with $(s>0)$, the difference between its scattering state and its initial datum belongs to $(mathcal H^{s+sigma})$ for every $(0<sigma<min{2s,1})$. The proof combines the lens transform, which turns the large-time problem into a finite-time endpoint problem, with bilinear Strichartz estimates and harmonic spectral analysis. The emphasis will be on the mechanism behind the regularity gain.
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