JournalsjemsVol. 17, No. 7pp. 1687–1759

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

  • Andrea R. Nahmod

    University of Massachusetts, Amherst, USA
  • Gigliola Staffilani

    Massachusetts Institute of Technology, Cambridge, USA
Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space cover
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Abstract

In this paper 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 H1(T3)H^1(\mathbb T^3).

We also prove a long time existence result; more precisely we prove that for fixed T>0T>0 there exists a set ΣT\Sigma_T, P(ΣT)>0\mathbb P(\Sigma_T) > 0 such that any data ϕω(x)Hγ(T3),γ<1,ωΣT\phi^{\omega}(x) \in H^{\gamma}(\mathbb T^3), \gamma<1, \omega \in \Sigma_T, evolves up to time TT into a solution u(t)u(t) with u(t)eitΔϕωC([0,T];Hs(T3))u(t) - e^{it\Delta} \phi^{\omega} \in C([0,T]; H^s(\mathbb T^3)), s=s(γ)>1s=s(\gamma)>1. In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space H1(T3)H^1(\mathbb T^3), that is in the supercritical scaling regime.

Cite this article

Andrea R. Nahmod, Gigliola Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1687–1759

DOI 10.4171/JEMS/543