A remark on normal forms and the “upside-down” \(I\)-method for periodic NLS: growth of higher Sobolev norms. (English) Zbl 1310.35213
Summary: We study growth of higher Sobolev norms of solutions of the one-dimensional periodic nonlinear Schrödinger equation (NLS). By a combination of the normal form reduction and the upside-down \(I\)-method, we establish
\[
\| u(t)\|_{H^s}\lesssim(1+| t|)^{\alpha(s-1)+}
\]
with \(\alpha=1\) for a general power nonlinearity. In the quintic case, we obtain the above estimate with \(\alpha=1/2\) via the space-time estimate due to J. Bourgain [Ergodic Theory Dyn. Syst. 24, No. 5, 1331–1357 (2004; Zbl 1087.37056); J. Anal. Math. 94, 125–157 (2004; Zbl 1084.35085)]. In the cubic case, we compute concretely the terms arising in the first few steps of the normal form reduction and prove the above estimate with \(\alpha=4/9\). These results improve the previously known results (except for the quintic case). In the Appendix, we also show how Bourgain’s idea in [Zbl 1087.37056] on the normal form reduction for the quintic nonlinearity can be applied to other powers.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B45 | A priori estimates in context of PDEs |
| 35B10 | Periodic solutions to PDEs |
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