Analysis & PDE

  • Anal. PDE
  • Volume 11, Number 8 (2018), 1841-1879.

Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation

Mouhamadou Sy

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/apde.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The Benjamin–Ono equation describes the propagation of internal waves in a stratified fluid. In the present work, we study large time dynamics of its regular solutions via some probabilistic point of view. We prove the existence of an invariant measure concentrated on C(T) and establish some qualitative properties of this measure. We then deduce a recurrence property of regular solutions and other corollaries using ergodic theorems. The approach used in this paper applies to other equations with infinitely many conservation laws, such as the KdV and cubic Schrödinger equations in one dimension. It uses the fluctuation-dissipation-limit approach and relies on a uniform smoothing lemma for stationary solutions to the damped-driven Benjamin–Ono equation.

Article information

Source
Anal. PDE, Volume 11, Number 8 (2018), 1841-1879.

Dates
Received: 14 November 2016
Revised: 10 January 2018
Accepted: 14 February 2018
First available in Project Euclid: 15 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.apde/1547521440

Digital Object Identifier
doi:10.2140/apde.2018.11.1841

Mathematical Reviews number (MathSciNet)
MR3812859

Zentralblatt MATH identifier
1388.35175

Subjects
Primary: 35A09: Classical solutions 35B40: Asymptotic behavior of solutions 35Q51: Soliton-like equations [See also 37K40] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)

Keywords
Benjamin–Ono equation invariant measure long time behavior regular solutions inviscid limit

Citation

Sy, Mouhamadou. Invariant measure and long time behavior of regular solutions of the Benjamin–Ono equation. Anal. PDE 11 (2018), no. 8, 1841--1879. doi:10.2140/apde.2018.11.1841. https://projecteuclid.org/euclid.apde/1547521440


Export citation

References

  • L. Abdelouhab, J. L. Bona, M. Felland, and J.-C. Saut, “Nonlocal models for nonlinear, dispersive waves”, Phys. D 40:3 (1989), 360–392.
  • Y. Deng, “Invariance of the Gibbs measure for the Benjamin–Ono equation”, J. Eur. Math. Soc. $($JEMS$)$ 17:5 (2015), 1107–1198.
  • Y. Deng, N. Tzvetkov, and N. Visciglia, “Invariant measures and long time behaviour for the Benjamin–Ono equation, III”, Comm. Math. Phys. 339:3 (2015), 815–857.
  • I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics 113, Springer, 1991.
  • S. B. Kuksin, “The Eulerian limit for 2D statistical hydrodynamics”, J. Statist. Phys. 115:1-2 (2004), 469–492.
  • S. B. Kuksin, “On distribution of energy and vorticity for solutions of 2D Navier–Stokes equation with small viscosity”, Comm. Math. Phys. 284:2 (2008), 407–424.
  • S. Kuksin and A. Shirikyan, “Randomly forced CGL equation: stationary measures and the inviscid limit”, J. Phys. A 37:12 (2004), 3805–3822.
  • S. Kuksin and A. Shirikyan, Mathematics of two-dimensional turbulence, Cambridge Tracts in Mathematics 194, Cambridge University Press, 2012.
  • Y. Matsuno, Bilinear transformation method, Mathematics in Science and Engineering 174, Academic Press, Orlando, FL, 1984.
  • L. Molinet, “Global well-posedness in $L^2$ for the periodic Benjamin–Ono equation”, Amer. J. Math. 130:3 (2008), 635–683.
  • L. Molinet and D. Pilod, “The Cauchy problem for the Benjamin–Ono equation in $L^2$ revisited”, Anal. PDE 5:2 (2012), 365–395.
  • A. Shirikyan, “Local times for solutions of the complex Ginzburg–Landau equation and the inviscid limit”, J. Math. Anal. Appl. 384:1 (2011), 130–137.
  • N. Tzvetkov and N. Visciglia, “Gaussian measures associated to the higher order conservation laws of the Benjamin–Ono equation”, Ann. Sci. Éc. Norm. Supér. $(4)$ 46:2 (2013), 249–299.
  • N. Tzvetkov and N. Visciglia, “Invariant measures and long-time behavior for the Benjamin–Ono equation”, Int. Math. Res. Not. 2014:17 (2014), 4679–4714.
  • N. Tzvetkov and N. Visciglia, “Invariant measures and long time behaviour for the Benjamin–Ono equation, II”, J. Math. Pures Appl. $(9)$ 103:1 (2015), 102–141.
  • P. E. Zhidkov, Korteweg–de Vries and nonlinear Schrödinger equations: qualitative theory, Lecture Notes in Mathematics 1756, Springer, 2001.
  • P. E. Zhidkov, “On an infinite sequence of invariant measures for the cubic nonlinear Schrödinger equation”, Int. J. Math. Math. Sci. 28:7 (2001), 375–394.