On the Kuramoto-Sivashinsky equation in a disk

Vladimir Varlamov

Annales Polonici Mathematici (2000)

  • Volume: 73, Issue: 3, page 227-256
  • ISSN: 0066-2216

Abstract

top
We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion.

How to cite

top

Varlamov, Vladimir. "On the Kuramoto-Sivashinsky equation in a disk." Annales Polonici Mathematici 73.3 (2000): 227-256. <http://eudml.org/doc/262852>.

@article{Varlamov2000,
abstract = {We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion.},
author = {Varlamov, Vladimir},
journal = {Annales Polonici Mathematici},
keywords = {first initial-boundary value problem; long-time asymptotics; Kuramoto-Sivashinsky equation; disk; uniform in space long-time asymptotic expansion; existence and the uniqueness of a global in time solution; small initial data},
language = {eng},
number = {3},
pages = {227-256},
title = {On the Kuramoto-Sivashinsky equation in a disk},
url = {http://eudml.org/doc/262852},
volume = {73},
year = {2000},
}

TY - JOUR
AU - Varlamov, Vladimir
TI - On the Kuramoto-Sivashinsky equation in a disk
JO - Annales Polonici Mathematici
PY - 2000
VL - 73
IS - 3
SP - 227
EP - 256
AB - We consider the first initial-boundary value problem for the 2-D Kuramoto-Sivashinsky equation in a unit disk with homogeneous boundary conditions, periodicity conditions in the angle, and small initial data. Apart from proving the existence and uniqueness of a global in time solution, we construct it in the form of a series in a small parameter present in the initial conditions. In the stable case we also obtain the uniform in space long-time asymptotic expansion of the constructed solution and its asymptotics with respect to the nonlinearity constant. The method can work for other dissipative parabolic equations with dispersion.
LA - eng
KW - first initial-boundary value problem; long-time asymptotics; Kuramoto-Sivashinsky equation; disk; uniform in space long-time asymptotic expansion; existence and the uniqueness of a global in time solution; small initial data
UR - http://eudml.org/doc/262852
ER -

References

top
  1. [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and the Inverse Scattering, London Math. Soc., Cambridge Univ. Press, 1991. Zbl0762.35001
  2. [2] D. Armbruster, G. Guckenheimer, and P. Holmes, Kuramoto-Sivashinsky dynamics on the center unstable manifold, SIAM J. Appl. Math. 49 (1989), 676-691. Zbl0687.34036
  3. [3] N. G. Berloff and L. N. Howard, Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. Appl. Math. 99 (1997), 1-24. Zbl0880.35105
  4. [4] D. J. Berney, Long waves on liquid films, J. Math. Phys. 45 (1966), 150-155. Zbl0148.23003
  5. [5] K. Chandrasekharan, Elliptic Functions, Springer, Berlin, 1985. 
  6. [6] P. Collet, J. P. Eckmann, H. Epstein, and J. Stubbe, A global attracting set for the Kuramoto-Sivashinsky equation, Comm. Math. Phys. 152 (1993), 203-214. Zbl0777.35073
  7. [7] C. Foiaş, B. Nicolaenko, G. R. Sell, and R. Temam, Inertial manifolds for dissipative partial differential equations, C. R. Acad. Sci. Paris Sér. I 301 (1995), 199-241. 
  8. [8] C. Foiaş, B. Nicolaenko, G. R. Sell, and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl. 67 (1988), 197-226. Zbl0694.35028
  9. [9] A. S. Fokas, Initial-boundary value problems for soliton equations, in: Proc. III Potsdam-V Kiev Internat. Workshop 1991, Springer, Berlin, 1992. 
  10. [10] A. S. Fokas and A. R. Its, Soliton generation for initial-boundary value problems, Phys. Rev. Lett. 68 (1992), 3117-3120. Zbl0969.35537
  11. [11] A. S. Fokas and A. R. Its, The linearization of the initial-boundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 27 (1996), 738-764. Zbl0851.35122
  12. [12] J. Goodman, Stability of the Kuramoto-Sivashinsky equation and related systems, Comm. Pure Appl. Math. 47 (1994), 293-306. Zbl0809.35105
  13. [13] A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids 28 (1985), 37-45. Zbl0565.76051
  14. [14] E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, 6th ed., Teubner, Stuttgart, 1960. Zbl0087.12801
  15. [15] Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from the thermal equilibrium, Progr. Theoret. Phys. 55 (1976), 356-369. 
  16. [16] Y. Kuramoto and T. Yamada, Turbulent state in chemical reactions, ibid. 56 (1976), 679. 
  17. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, New York, 1972. 
  18. [18] D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D 19 (1986), 89-111. Zbl0603.35080
  19. [19] D. Michelson, Bunsen flames as steady solutions of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 23 (1991), 364-386. Zbl0787.34019
  20. [20] D. Michelson, Stability of the Bunsen flame profiles in the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 27 (1996), 765-781. Zbl0851.34020
  21. [21] L. Molinet, Local dissipativity in L₂ for the Kuramoto-Sivashinsky equation in spatial dimension two, J. Dynam. Differential Equations, to appear. Zbl0970.35132
  22. [22] P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, Transl. Math. Monographs 133, Amer. Math. Soc., Providence, RI, 1994. Zbl0802.35002
  23. [23] B. Nicolaenko and B. Scheurer, Remarks on the Kuramoto-Sivashinsky equation, Phys. D 12 (1984), 391-395. Zbl0576.35058
  24. [24] B. Nicolaenko, B. Scheurer, and R. Temam, Some global dynamical properties of the Kuramoto-Sivashinsky equation: nonlinear stability and attractors, ibid. 16 (1985), 155-183. Zbl0592.35013
  25. [25] F. W. J. Olver, Introduction to Asymptotics and Special Functions, Acad. Press, New York, 1974. Zbl0308.41023
  26. [26] S. V. Raghavan, J. B. McLeod, and W. O. Troy, A singular perturbation problem arising from the Kuramoto-Sivashinsky equation, Differential Integral Equations 19 (1997), 1-36. Zbl0879.34028
  27. [27] G. Raugel and G. Sell, Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), 503-568. Zbl0787.34039
  28. [28] G. R. Sell and M. Taboada, Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains, Nonlinear Anal. 18 (1992), 671-687. Zbl0784.35046
  29. [29] G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math. 39 (1980), 67-82. Zbl0464.76055
  30. [30] G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Ann. Rev. Fluid Mech. 15 (1983), 179-199. Zbl0538.76053
  31. [31] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1958. Zbl0097.27601
  32. [32] G. Tolstov, Fourier Series, Dover, New York, 1962. 
  33. [33] J. Topper and T. Kawahara, Approximate equations for nonlinear waves on a viscous fluid, J. Phys. Soc. Japan 44 (1978), 663-666. Zbl1334.76054
  34. [34] W. C. Troy, The existence of steady solutions of the Kuramoto-Sivashinsky equation, J. Differential Equations 82 (1989), 269-313. Zbl0693.34053
  35. [35] V. V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), 619-634. Zbl0844.35095
  36. [36] V. V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, ibid. 10 (1997), 1197-1211. Zbl0940.35162
  37. [37] V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dynam. Systems 4 (1998), 431-444. Zbl0952.35103
  38. [38] V. V. Varlamov, Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation, Abstract Appl. Anal. 2 (1998), 97-115. 
  39. [39] V. V. Varlamov, The third-order nonlinear evolution equation governing wave propagation in relaxing media, Stud. Appl. Math. 99 (1997), 25-47. Zbl0881.35079
  40. [40] V. V. Varlamov, On the damped Boussinesq equation in a circle, Nonlinear Anal. 38 (1999), 447-470. Zbl0938.35146
  41. [41] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, London, 1966. Zbl0174.36202

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.