Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space
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1308.1169.pdf
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Author(s)
Nahmod, Andrea
Staffilani, Gigliola
Date Issued
2015
Journal
Journal of the European Mathematical Society
Publisher
European Mathematical Publishing House
Citation
Nahmod, Andrea and Gigliola Staffilani. “Almost Sure Well-Posedness for the Periodic 3D Quintic Nonlinear Schrödinger Equation Below the Energy Space.” Journal of the European Mathematical Society 17, 7 (2015): 1687–1759 © 2015 European Mathematical Society
Version
Author's final manuscript
Abstract
We prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is, below the critical space H ¹ (T³). We also prove a long time existence result; more precisely, we show that for fixed T > 0 there exists a set ∑T with P(∑T ) > 0 such that any data Φ[superscript ω] (x) ∈ Ha[superscript γ] (T³), γ < 1, ω ∈ ∑ [subscript T], evolves up to time T into a solution u(t) with u(t) - e [superscript itΔ]Φ[superscript ω] ∈ C([0, T]; H [superscript s](T³ )), s = s(γ) > 1. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H¹ (T³), that is, in the supercritical scaling regime. Keywords: Supercritical nonlinear Schrödinger equation; almost sure well-posedness; random data
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.4171/JEMS/543
