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Long time behavior of solutions for a damped Benjamin–Ono equation

Abstract

We consider the Benjamin–Ono equation on the torus with an additional damping term on the smallest Fourier modes (\(\cos \) and \(\sin \)). We first prove global well-posedness of this equation in \(L^2_{r,0}(\mathbb {T})\). Then, we describe the weak limit points of the trajectories in \(L^2_{r,0}(\mathbb {T})\) when time goes to infinity, and show that these weak limit points are strong limit points. Finally, we prove the boundedness of higher-order Sobolev norms for this equation. Our key tool is the Birkhoff map for the Benjamin–Ono equation, that we use as an adapted nonlinear Fourier transform.

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Acknowledgements

The author would like to thank her PhD advisor Patrick Gérard who introduced her to this problem and provided generous advice and encouragement. She also warmly thanks Christian Klein for valuable discussions and numerical simulations.

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Correspondence to Louise Gassot.

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Gassot, L. Long time behavior of solutions for a damped Benjamin–Ono equation. Math. Z. (2021). https://doi.org/10.1007/s00209-021-02849-w

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Keywords

  • Benjamin–Ono equation
  • Damping
  • Long time behavior
  • Birkhoff coordinates

Mathematics Subject Classification

  • 37K10
  • 35Q53