Exponential stability of the flow for a generalized Burgers equation on a circle

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Abstract

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the L1 norm. As a consequence, it is proved that the equation with a bounded external force possesses a unique bounded solution on R, which is exponentially stable in H1 as t→+∞. In the case of a random external force, we show that the difference between two trajectories goes to zero with probability 1.

About the authors

A. Djurdjevac

Freie Universitat Berlin

Author for correspondence.
Email: adjurdjevac@zedat.fu-berlin.de
Berlin, Germany

A. R. Shirikyan

CY Cergy Paris University; RUDN University

Email: Armen.Shirikyan@cyu.fr
Cergy-Pontoise, France; Moscow, Russia

References

  1. Бесов О. В., Ильин В. П., Никольский С. М. Интегральные представления функций и теоремы вложения. - М.: Наука, 1975.
  2. Kружков С. Н. О задаче Коши для некоторых классов квазилинейных параболических уравнений// Мат. заметки. - 1969. - 6, № 3. - С. 295-300.
  3. Крылов Н. В. Нелинейные эллиптические и параболические уравнения второго порядка. - М.: Наука, 1985.
  4. Крылов Н. В., Сафонов М. В. Некоторое свойство решений параболических уравнений с измеримыми коэффициентами// Изв. АН СССР. Сер. мат. - 1980. - 44, № 1. - С. 161-175.
  5. Ландис E. M. Уравнения второго порядка эллиптического и параболического типов. - М.: Наука, 1971.
  6. Лионс Ж.-Л. Некоторые методы решения нелинейных краевых задач. - М.: Мир, 1972.
  7. Лионс Ж.-Л., Мадженес Э. Неоднородные граничные задачи и их приложения. Т. 1. - М.: Мир, 1971.
  8. Bakhtin Y., Li L. Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation// Commun. Pure Appl. Math. - 2019. - 72, № 3. - С. 536-619.
  9. Boritchev A. Sharp estimates for turbulence in white-forced generalised Burgers equation// Geom. Funct. Anal. - 2013. - 23, № 6. - С. 1730-1771.
  10. Chung J., Kwon O. Asymptotic behavior for the viscous Burgers equation with a stationary source// J. Math. Phys. - 2016. - 57, № 10. - 101506.
  11. Dunlap A., Graham C., Ryzhik L. Stationary solutions to the stochastic Burgers equation on the line// Commun. Math. Phys. - 2021. - 382, № 2. - С. 875-949.
  12. Djurdjevac A., Rosati T. Synchronisation for scalar conservation laws via Dirichlet boundary// ArXiv. - 2022. - 2211.05814.
  13. Djurdjevac A., Shirikyan A. Stabilisation of a viscous conservation law by a one-dimensional external force// ArXiv. - 2022. - 2204.03427.
  14. Evans L. C. Partial differential equations. - Providence: Am. Math. Soc., 2010.
  15. Hill A. T., Su¨li E. Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains// Proc. Roy. Soc. Edinburgh Sect. A. - 1995. - 125, № 2. - С. 439-448.
  16. H¨ormander L. Lectures on nonlinear hyperbolic differential equations. - Berlin: Springer, 1997.
  17. Jauslin H. R., Kreiss H. O., Moser J. On the forced Burgers equation with periodic boundary conditions// В сб.: «Differential equations: La Pietra 1996». - Providence: Am. Math. Soc., 1999. - С. 133-153.
  18. Kalita P., Zgliczyn´ski P. On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions// Proc. Roy. Soc. Edinburgh Sect. A. - 2020. - 150, № 4. - С. 2025-2054.
  19. Kifer Y. The Burgers equation with a random force and a general model for directed polymers in random environments// Probab. Theory Related Fields. - 1997. - 108, № 1. - С. 29-65.
  20. Shirikyan A. Global exponential stabilisation for the Burgers equation with localised control// J. E´ c. Polytech. Math. - 2017. - 4. - С. 613-632.
  21. Sina˘ı Ya. G. Two results concerning asymptotic behavior of solutions of the Burgers equation with force// J. Stat. Phys. - 1991. - 64, № 1-2. - С. 1-12.

Copyright (c) 2023 Djurdjevac A., Shirikyan A.R.

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