Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball I: The 2D case

Bourgain, Jean; Bulut, Aynur

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1267-1288.

Our first purpose is to extend the results from [14] on the radial defocusing NLS on the disc in ${ℝ}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs measure (which is the natural extension of the classical flow for smooth data). We follow a similar approach as in [8] exploiting certain additional a priori space–time bounds that are provided by the invariance of the Gibbs measure.Next, we consider the radial focusing equation with cubic nonlinearity (the mass-subcritical case was studied in [15]) where the Gibbs measure is subject to an $L^2$-norm restriction. A phase transition is established. For sufficiently small $L^2$-norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics. For sufficiently large $L^2$-norm cutoff, the Gibbs measure concentrates on delta functions centered at 0. This phenomenon is similar to the one observed in the work of Lebowitz, Rose, and Speer [13] on the torus.

[1] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal. 3 no. 2 (1993), 107 -156 | EuDML | MR | Zbl

[2] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal. 3 no. 3 (1993), 209 -262 | EuDML | MR | Zbl

[3] J. Bourgain, Periodic nonlinear Schrödinger equation in invariant measures, Commun. Math. Phys. 166 (1994), 1 -24 | MR | Zbl

[4] J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Commun. Math. Phys. 176 (1996), 421 -445 | MR | Zbl

[5] J. Bourgain, Invariant measures for the Gross–Piatevskii equation, J. Math. Pures Appl. 76 (1997), 649 -702 | MR | Zbl

[6] J. Bourgain, Nonlinear Schrödinger equations, Hyperbolic Equations and Frequency Interactions, IAS/Park City Math. Ser. vol. 5 , Amer. Math. Soc., Providence, RI (1999), 3 -157 | MR | Zbl

[7] J. Bourgain, A. Bulut, Gibbs measure evolution in radial nonlinear wave and Schrödinger equations on the ball, C. R. Math. 350 no. 11–12 (2012), 571 -575 | MR | Zbl

[8] J. Bourgain, A. Bulut, Invariant Gibbs measure evolution for the 3D radial NLW on the ball, preprint, 2012. | MR

[9] J. Bourgain, A. Bulut, Almost sure global well posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3D case, arXiv:1302.5409 (2013) | MR | Zbl

[10] D. Brydges, G. Slade, Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation, Commun. Math. Phys. 182 (1996), 485 -504 | MR | Zbl

[11] N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 no. 3 (2008), 449 -475 N. Burq, N. Tzvetkov, Random data Cauchy theory for supercritical wave equations. II. A global existence result, Invent. Math. 173 no. 3 (2008), 477 -496 | MR | Zbl

[12] O. Kavian, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. Am. Math. Soc. 299 no. 1 (1987), 193 -203 | MR | Zbl

[13] J. Lebowitz, R. Rose, E. Speer, Statistical mechanics of the nonlinear Schrödinger equation, J. Stat. Phys. 50 (1988), 657 -687 | MR | Zbl

[14] N. Tzvetkov, Invariant measures for the defocusing nonlinear Schrödinger equation, Ann. Inst. Fourier 58 no. 7 (2008), 2543 -2604 | EuDML | Numdam | MR | Zbl

[15] N. Tzvetkov, Invariant measures for the nonlinear Schrödinger equation on the disc, Dyn. Partial Differ. Equ. 3 no. 2 (2006), 111 -160 | MR | Zbl