Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation

1 Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, via E. Orabona 4, 70125 Bari, Italy
2 Dipartimento di Matematica, Universita di Bari, via E. Orabona 4, 70125 Bari, Italy

The Kuramoto-Sinelshchikov equation describes the evolution of a phase turbulence in reaction-diffusion systems or the evolution of the plane flame propagation, taking in account the combined influence of diffusion and thermal conduction of the gas on the stability of a plane flame front. In this paper, we prove the well-posedness of the classical solutions for the initial-boundary value problem for this equation, under appropriate boundary conditions.
  Figure/Table
  Supplementary
  Article Metrics

Keywords existence; uniqueness; stability; Kuramoto-Sinelshchikov type equation; initial-boundary value problem

Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo. On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation. Mathematics in Engineering, 2021, 3(4): 1-43. doi: 10.3934/mine.2021036

References

  • 1. Amiranashvili S, Vladimirov AG, Bandelow U (2008) Solitary-wave solutions for few-cycle optical pulses. Phys Rev A 77: 063821.    
  • 2. Amiranashvili S, Vladimirov AG, Bandelow U (2010) A model equation for ultrashort optical pulses around the zero dispersion frequency. Eur Phys J D 58: 219-226.    
  • 3. Armaou A, Christofides PD (2000) Feedback control of the Kuramoto-Sivashinsky equation. Phys D 137: 49-61.    
  • 4. Benney DJ (1966) Long waves on liquid films. J Math Phys 45: 150-155.    
  • 5. Biagioni HA, Bona JL, Iório Jr RJ, et al. (1996) On the Korteweg-de Vries-Kuramoto-Sivashinsky equation. Adv Differ Equ 1: 1-20.
  • 6. Cerpa E (2010) Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Commun Pure Appl Anal 9: 91-102.    
  • 7. Chen LH, Chang HC (1986) Nonlinear waves on liquid film surfaces-II. Bifurcation analyses of the long-wave equation. Chem Eng Sci 41: 2477-2486.
  • 8. Christofides PD, Armaou A (2000) Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control. Syst Control Lett 39: 283-294.    
  • 9. Coclite GM, di Ruvo L (2014) Convergence of the Ostrovsky equation to the Ostrovsky-Hunter one. J Differ Equations 256: 3245-3277.    
  • 10. Coclite GM, di Ruvo L (2015) Dispersive and diffusive limits for Ostrovsky-Hunter type equations. NoDEA Nonlinear Diff 22: 1733-1763.    
  • 11. Coclite GM, di Ruvo L (2015) Well-posedness of bounded solutions of the non-homogeneous initial-boundary value problem for the Ostrovsky-Hunter equation. J Hyperbolic Differ Equ 12: 221-248.    
  • 12. Coclite GM, di Ruvo L (2016) Convergence of the Kuramoto-Sinelshchikov equation to the Burgers one. Acta Appl Math 145: 89-113.    
  • 13. Coclite GM, di Ruvo L (2016) Convergence of the solutions on the generalized Korteweg-de Vries equation. Math Model Anal 21: 239-259.    
  • 14. Coclite GM, di Ruvo L (2017) A singular limit problem for conservation laws related to the Rosenau equation. J Abstr Differ Equ Appl 8: 24-47.
  • 15. Coclite GM, di Ruvo L (2019) Discontinuous solutions for the generalized short pulse equation. Evol Equ Control The 8: 737-753.    
  • 16. Coclite GM, di Ruvo L (2020) Convergence of the Rosenau-Korteweg-de Vries equation to the Korteweg-de Vries one. Contemporary Mathematics.
  • 17. Coclite GM, di Ruvo L (2020) A note on the non-homogeneous initial boundary problem for an Ostrovsky-Hunter1 type equation. Discrete Contin Dyn Syst Ser S 13: 3357-3389.
  • 18. Coclite GM, di Ruvo L (2020) On classical solutions for a Kuramoto-Sinelshchikov-Velarde-type equation. Algorithms 13: 77.    
  • 19. Coclite GM, di Ruvo L (2020) On the solutions for an Ostrovsky type equation. Nonlinear Anal Real 55: 103141.    
  • 20. Cohen BI, Krommes JA, Tang WM, et al. (1976) Non-linear saturation of the dissipative trappedion mode by mode coupling. Nucl Fusion 16: 971-992.    
  • 21. Foias C, Nicolaenko B, Sell GR, et al. (1988) Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension. J Math Pure Appl 67: 197-226.
  • 22. Giacomelli L, Otto F (2005) New bounds for the Kuramoto-Sivashinsky equation. Commun Pure Appl Math 58: 297-318.    
  • 23. Hooper AP, Grimshaw R (1985) Nonlinear instability at the interface between two viscous fluids. Phys Fluids 28: 37-45.    
  • 24. Hu C, Temam R (2001) Robust control of the Kuramoto-Sivashinsky equation. Dyn Contin Discrete Impuls Syst Ser B Appl Algorithms 8: 315-338.
  • 25. Kenig CE, Ponce G, Vega L (1993) Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun Pure Appl Math 46: 527-620.    
  • 26. Khalique C (2012) Exact solutions of the generalized kuramoto-sivashinsky equation. CJMS 1: 109-116.
  • 27. Korteweg DDJ, de Vries DG (1895) XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 39: 422-443.
  • 28. Kudryashov NA (1990) Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys Lett A 147: 287-291.    
  • 29. Kudryashov NA (2009) On "new travelling wave solutions" of the KdV and the KdV-Burgers equations. Commun Nonlinear Sci Numer Simul 14: 1891-1900.    
  • 30. Kuramoto Y (1978) Diffusion-induced chaos in reaction systems. Prog Theor Phys Supp 64: 346- 367.    
  • 31. Kuramoto Y, Tsuzuki T (1975) On the formation of dissipative structures in reaction-diffusion systems: Reductive perturbation approach. Prog Theor Phys 54: 687-699.    
  • 32. Kuramoto Y, Tsuzuki T (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog Theor Phys 55: 356-369.    
  • 33. LaQuey RE, Mahajan SM, Rutherford PH, et al. (1975) Nonlinear saturation of the trapped-ion mode. Phys Rev Lett 34: 391-394.    
  • 34. Leblond H, Mihalache D (2009) Few-optical-cycle solitons: Modified Korteweg-de vries sinegordon equation versus other non-slowly-varying-envelope-approximation models. Phys Rev A 79: 063835.    
  • 35. Leblond H, Mihalache D (2013) Models of few optical cycle solitons beyond the slowly varying envelope approximation. Phys Rep 523: 61-126.    
  • 36. Leblond H, Sanchez F (2003) Models for optical solitons in the two-cycle regime. Phys Rev A 67: 013804.    
  • 37. LeFloch PG, Natalini R (1999) Conservation laws with vanishing nonlinear diffusion and dispersion. Nonlinear Anal 36: 213-230.    
  • 38. Li C, Chen G, Zhao S (2004) Exact travelling wave solutions to the generalized kuramotosivashinsky equation. Lat Am Appl Res 34: 65-68.
  • 39. Li J, Zhang BY, Zhang Z (2017) A nonhomogeneous boundary value problem for the KuramotoSivashinsky equation in a quarter plane. Math Method Appl Sci 40: 5619-5641.    
  • 40. Li J, Zhang BY, Zhang Z (2020) A non-homogeneous boundary value problem for the KuramotoSivashinsky equation posed in a finite interval. ESAIM Control Optim Calc Var 26: 43.    
  • 41. Lin SP (1974) Finite amplitude side-band stability of a viscous film. J Fluid Mech 63: 417-429.    
  • 42. Liu WJ, Krstić M (2001) Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation. Nonlinear Anal 43: 485-507.    
  • 43. Nicolaenko B, Scheurer B, Temam R (1985) Some global dynamical properties of the kuramotosivashinsky equations: Nonlinear stability and attractors. Physica D 16: 155-183.    
  • 44. Nicolaenko B, Scheurer B (1984) Remarks on the kuramoto-sivashinsky equation. Physica D 12: 391-395.    
  • 45. Sajjadian M (2014) The shock profile wave propagation of Kuramoto-Sivashinsky equation and solitonic solutions of generalized Kuramoto-Sivashinsky equation. Acta Univ Apulensis Math Inform 38: 163-176.
  • 46. Schonbek ME (1982) Convergence of solutions to nonlinear dispersive equations. Commun Part Diff Eq 7: 959-1000.    
  • 47. Sivashinsky G (1977) Nonlinear analysis of hydrodynamic instability in laminar flamesâ-I. Derivation of basic equations. Acta Astronaut 4: 1177-1206.
  • 48. Tadmor E (1986) The well-posedness of the Kuramoto-Sivashinsky equation. SIAM J Math Anal 17: 884-893.    
  • 49. Taylor ME (2011) Partial Differential Equations I. Basic Theory, 2 Eds., New York: Springer.
  • 50. Topper J, Kawahara T (1978) Approximate equations for long nonlinear waves on a viscous fluid. J Phys Soc JPN 44: 663-666.    
  • 51. Xie Y (2013) Solving the generalized Benney equation by a combination method. Int J Nonlinear Sci 15: 350-354.

 

Reader Comments

your name: *   your email: *  

© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved