Out-of-equilibrium dynamics and statistics of dispersive PDE

Hani, Zaher

Journées équations aux dérivées partielles (2016), Exposé no. 5, 12 p.

The purpose of this note is to report on some recent advances in the study of out-of-equilibrium behavior of dispersive PDE. One can address this problematic from two different perspectives: a dynamical systems one, and a statistical physics one. The dynamical systems perspective corresponds to constructing solutions exhibiting “energy cascade” between scales, whereas the statistical physics perspective corresponds to deriving effective equations for the dynamics under some “macroscopic limits” in what is often called wave turbulence theory. The rigorous justification of this theory is an outstanding open problem from a rigorous mathematical point of view, and we will touch on it here. We shall discuss some recent attempts to better understand both of the above perspectives.

Mots clés : Modified scattering, nonlinear Schrödinger equation, wave guide manifolds, energy cascade, weak turbulence

[1] J. Bourgain, Aspects of long time behaviour of solutions of nonlinear Hamiltonian evolution equations, Geom. Funct. Anal. 5 (1995), no. 2, 105–140.

[2] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 1996, no. 6, 277–304.

[3] J. Bourgain, On growth in time of Sobolev norms of smooth solutions of nonlinear Schrödinger equations in ${ℝ}^D$. J. Anal. Math. 72 (1997), 299–310.

[4] J. Bourgain, Problems in Hamiltonian PDE’s, Geom. Funct. Anal. 2000, Special Volume, Part I, 32–56.

[5] T. Buckmaster, P. Germain, Z. Hani, and J. Shatah, Effective dynamics of nonlinear Schrödinger equations on large domains. Preprint.

[6] T. Buckmaster, P. Germain, Z. Hani, and J. Shatah, Analysis of the (CR) equation in higher dimensions. Preprint.

[7] R. Carles, E. Faou, Energy cascades for NLS on ${𝕋}^d$, Discrete Contin. Dyn. Syst. 32 (2012) 2063–2077.

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), 39–113.

[9] Weinan E., K. Khanin, A. Mazel, Y. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math., 151 (3), 877–960 (2000).

[10] J.-P. Eckmann, C.-A. Pillet, L. Rey-Bellet, L., Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Comm. Math. Phys. 201 (1999), no. 3, 657–697.

[11] E. Faou, P. Germain, and Z. Hani, The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation, Journal of the AMS (JAMS). Published electronically: October 20, 2015 (68 pages).

[12] J. Ford, The Fermi-Pasta-Ulam Problem: Paradox turned discovery. Physics Reports (Review Section of Physics Letters) 213, No. 5(1992) 271–310. North-Holland.

[13] I. Gallagher, L. Saint-Raymond, B. Texier, From Newton to Boltzmann: hard spheres and short-range potentials. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2013.

[14] G. Galavotti, The Fermi-Pasta-Ulam Problem: a status report, Lecture Notes in Physics 728, Springer 2008.

[15] P. Gerard, S. Grellier, The cubic Szegö equation and Hankel operators. Preprint arXiv:1508.06814.

[16] P. Germain, Z. Hani, and L. Thomann, On the continuous resonant equation for NLS. Part I. Deterministic analysis. J. Math. Pures Appl. (9) 105 (2016), no. 1, 131–163.

[17] P. Germain, Z. Hani, and L. Thomann, On the continuous resonant equation for NLS. Part II. Probabilistic analysis. Analysis & PDE 8-7 (2015), 1733–1756.

[18] M. Guardia and V. Kaloshin, Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation, Journal of the Eur. Math. Society, 17 (1): 71–149 (2015).

[19] M. Guardia, E. Haus and M. Procesi, Growth of Sobolev norms for the analytic NLS on ${𝕋}^2$, Advances in Mathematics, published online, 2016.

[20] M. Hairer, How hot can a heat bath get?, Comm. Math. Phys. 292 (2009), no. 1, 131–177.

[21] M. Hairer, J. Mattingly, Slow energy dissipation in anharmonic oscillator chains. Comm. Pure Appl. Math. 62 (2009), no. 8, 999–1032.

[22] M. Hairer, J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006), no. 3, 993–1032.

[23] Z. Hani, Global and dynamical aspects of the nonlinear Schrödinger equations on compact manifolds, UCLA Ph.D Thesis, 2011.

[24] Z. Hani, Long-time strong instability and unbounded orbits for some nonlinear Schrödinger equations on ${𝕋}^2$, Archive for Rational Mechanics and Analysis, 2014, Volume 211, Issue 3, pp 929–964.

[25] Z. Hani and B. Pausader, On scattering for the quintic defocusing nonlinear Schrödinger equation on $ℝ\times {𝕋}^2$, Communications on Pure and Applied Mathematics, volume 67, Issue 9, pages 1466–1542, 2014.

[26] Z. Hani, B. Pausader, N. Tzvetkov, N. Visciglia, Modified scattering for the cubic Schrödinger equation on product spaces and applications, Forum of Math. Pi, Volume 3 / 2015, e4 (63 pages).

[27] Z. Hani, B. Pausader, N. Tzvetkov, N. Visciglia, Growing Sobolev norms for the cubic defocusing Schroedinger equation. Seminaire Laurent Schwartz - EDP et applications (2013-2014), Exp. No. 16, 11 p.

[28] Z. Hani and L. Thomann, Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping. To appear in Comm. Pure and Applied Math (CPAM).

[29] E. Haus and M. Procesi, Growth of Sobolev norms for the quintic NLS on ${𝕋}^2$, Analysis and Partial Differential Equations, 8 (4), 883–922, 2015.

[30] D.R. Heath-Brown, A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481, (1996) 149–206.

[31] D.R. Heath-Brown, Analytic Methods For The Distribution of Rational Points On Algebraic Varieties in Equidistribution in Number Theory, An Introduction. NATO Science Series, 2007.

[32] S. B. Kuksin, Oscillations in space-periodic nonlinear Schrödinger equations, Geom. Funct. Anal. 7 (1997), no. 2, 338–363.

[33] O. Lanford, On the derivation of the Boltzmann equation. Astérisque 40, 117–137 (1976).

[34] A. Majda, Introduction to PDEs and waves for the atmosphere and ocean. Courant Lecture Notes in Mathematics, 9. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.

[35] A. Majda, D. McLaughlin, E. Tabak, A one-dimensional model for dispersive wave turbulence., J. Nonlinear Sci. 7 (1997), no. 1, 9–44.

[36] S. Nazarenko, Wave Turbulence, Lecture Notes in Physics 825 Springer.

[37] R. Peierls, The kinetic theory of thermal conduction in crystals, Annalen der Physik 3 (8): 1055–1101 (1929).

[38] L. Rey-Bellet, Open classical systems. Open quantum systems. II, 41–78, Lecture Notes in Math., 1881, Springer, Berlin, 2006.

[39] Y. Sinai, Hyperbolic billiards. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 249–260, Math. Soc. Japan, Tokyo, 1991.

[40] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger Equations on $S^1$, Differential Integral Equations 24 (2011), no. 7-8, 653–718.

[41] G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J. 86 (1997), no. 1, 109–142.

[42] C. Sulem and P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999.

[43] T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics, 106, American Mathematical Society, Providence, RI, 2006.

[44] N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces, Comm. Partial Differential Equations, vol. 37, 2012, n.1, pp. 125–135.