## Analysis & PDE

• Anal. PDE
• Volume 8, Number 7 (2015), 1733-1756.

### On the continuous resonant equation for NLS, II: Statistical study

#### Abstract

We consider the continuous resonant (CR) system of the 2-dimensional cubic nonlinear Schrödinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of NLS in confined regimes (e.g., on a compact domain or with a trapping potential). The system was derived and studied from a deterministic viewpoint in several earlier works, which uncovered many of its striking properties. This manuscript is devoted to a probabilistic study of this system. Most notably, we construct global solutions in negative Sobolev spaces, which leave Gibbs and white noise measures invariant. Invariance of white noise measure seems particularly interesting in view of the absence of similar results for NLS.

#### Article information

Source
Anal. PDE, Volume 8, Number 7 (2015), 1733-1756.

Dates
Received: 15 April 2015
Revised: 4 June 2015
Accepted: 29 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843170

Digital Object Identifier
doi:10.2140/apde.2015.8.1733

Mathematical Reviews number (MathSciNet)
MR3399137

Zentralblatt MATH identifier
1326.35344

#### Citation

Germain, Pierre; Hani, Zaher; Thomann, Laurent. On the continuous resonant equation for NLS, II: Statistical study. Anal. PDE 8 (2015), no. 7, 1733--1756. doi:10.2140/apde.2015.8.1733. https://projecteuclid.org/euclid.apde/1510843170

#### References

• A. Aftalion, X. Blanc, and F. Nier, “Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates”, J. Funct. Anal. 241:2 (2006), 661–702.
• S. Albeverio and A. B. Cruzeiro, “Global flows with invariant (Gibbs) measures for Euler and Navier–Stokes two-dimensional fluids”, Comm. Math. Phys. 129:3 (1990), 431–444.
• N. Burq and N. Tzvetkov, “Random data Cauchy theory for supercritical wave equations, I: Local theory”, Invent. Math. 173:3 (2008), 449–475.
• N. Burq, L. Thomann, and N. Tzvetkov, “Long time dynamics for the one dimensional non linear Schrödinger equation”, Ann. Inst. Fourier $($Grenoble$)$ 63:6 (2013), 2137–2198.
• N. Burq, L. Thomann, and N. Tzvetkov, “Remarks on the Gibbs measures for nonlinear dispersive equations”, preprint, 2014.
• J. Colliander and T. Oh, “Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L\sp 2(\mathbb{T})$”, Duke Math. J. 161:3 (2012), 367–414.
• G. Da Prato and A. Debussche, “Two-dimensional Navier–Stokes equations driven by a space-time white noise”, J. Funct. Anal. 196:1 (2002), 180–210.
• Y. Deng, “Two-dimensional nonlinear Schrödinger equation with random radial data”, Anal. PDE 5:5 (2012), 913–960.
• E. Faou, P. Germain, and Z. Hani, “The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation”, preprint, 2013. To appear in J. Amer. Math. Soc.
• P. Gérard, P. Germain, and L. Thomann, “On the lowest level Landau equation”, in preparation.
• P. Germain, Z. Hani, and L. Thomann, “On the continuous resonant equation for NLS, I: Deterministic analysis”, preprint, 2015. To appear in J. Math. Pures Appl. $(9)$.
• Z. Hani and L. Thomann, “Asymptotic behavior of the nonlinear Schrödinger with harmonic trapping”, Comm. Pure Appl. Math. (online publication July 2015).
• T. Hida, Brownian motion, Applications of Mathematics 11, Springer, New York, 1980.
• R. Imekraz, D. Robert, and L. Thomann, “On random Hermite series”, Trans. Amer. Math. Soc. (online publication April 2015).
• S. Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics 129, Cambridge University Press, Cambridge, 1997.
• H. Koch and D. Tataru, “$L\sp p$ eigenfunction bounds for the Hermite operator”, Duke Math. J. 128:2 (2005), 369–392.
• F. Nier, “Bose–Einstein condensates in the lowest Landau level: Hamiltonian dynamics”, Rev. Math. Phys. 19:1 (2007), 101–130.
• A. Poiret, “Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions”, preprint, 2012.
• A. Poiret, “Solutions globales pour l'équation de Schrödinger cubique en dimension 3”, preprint, 2012.
• A. Poiret, D. Robert, and L. Thomann, “Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator”, Anal. PDE 7:4 (2014), 997–1026.
• A. Poiret, D. Robert, and L. Thomann, “Random-weighted Sobolev inequalities on $\mathbb{R}\sp d$ and application to Hermite functions”, Ann. Henri Poincaré 16:2 (2015), 651–689.
• D. Robert and L. Thomann, “On random weighted Sobolev inequalities on $\R^{d}$ and applications”, pp. 115–135 in Spectral Theory and Partial Differential Equations, edited by G. Eskin et al., Contemporary Mathematics 640, Amer. Math. Soc., Providence, RI, 2015.
• A.-S. de Suzzoni, “Invariant measure for the cubic wave equation on the unit ball of $\mathbb{R}\sp 3$”, Dyn. Partial Differ. Equ. 8:2 (2011), 127–147.
• M. E. Taylor, Tools for PDE: pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs 81, Amer. Math. Soc., Providence, RI, 2000.
• L. Thomann, “Random data Cauchy problem for supercritical Schrödinger equations”, Ann. Inst. H. Poincaré Anal. Non Linéaire 26:6 (2009), 2385–2402.
• L. Thomann and N. Tzvetkov, “Gibbs measure for the periodic derivative nonlinear Schrödinger equation”, Nonlinearity 23:11 (2010), 2771–2791.
• K. Yajima and G. Zhang, “Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity”, J. Differential Equations 202:1 (2004), 81–110.