Scattering and Smoothing for the Two-Dimensional Cubic Nonlinear Schrödinger Equation
par
Centre de conférences marilyn et James Simons
IHES
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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Zhongkai Tao, Jian Wang