Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space
Author(s)
Nahmod, Andrea; Staffilani, Gigliola
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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
Date issued
2015Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
1435-9855
1435-9863