On the Kuramoto-Sivashinsky equation in a disk

• Autorzy: Vladimir Varlamov
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

first initial-boundary value problem; long-time asymptotics; Kuramoto-Sivashinsky equation; disk

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 73
• Numer: 3
• Strony: 227-256

Daty

wydano

2000

otrzymano

1999-09-07

Twórcy

autor

Vladimir Varlamov

• Departamento de Matemáticas, Escuela Colombiana de Ingenierí a, A.A. 14520, Bogotá, Colombia

Bibliografia

• [1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations, and the Inverse Scattering, London Math. Soc., Cambridge Univ. Press, 1991.
• [2] D. Armbruster, G. Guckenheimer, and P. Holmes, Kuramoto-Sivashinsky dynamics on the center unstable manifold, SIAM J. Appl. Math. 49 (1989), 676-691.
• [3] N. G. Berloff and L. N. Howard, Solitary and periodic solutions of nonlinear nonintegrable equations, Stud. Appl. Math. 99 (1997), 1-24.
• [4] D. J. Berney, Long waves on liquid films, J. Math. Phys. 45 (1966), 150-155.
• [5] K. Chandrasekharan, Elliptic Functions, Springer, Berlin, 1985.
• [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.
• [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] 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.
• [9] A. S. Fokas, Initial-boundary value problems for soliton equations, in: Proc. III Potsdam-V Kiev Internat. Workshop 1991, Springer, Berlin, 1992.
• [10] A. S. Fokas and A. R. Its, Soliton generation for initial-boundary value problems, Phys. Rev. Lett. 68 (1992), 3117-3120.
• [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.
• [12] J. Goodman, Stability of the Kuramoto-Sivashinsky equation and related systems, Comm. Pure Appl. Math. 47 (1994), 293-306.
• [13] A. P. Hooper and R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids 28 (1985), 37-45.
• [14] E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, 6th ed., Teubner, Stuttgart, 1960.
• [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] Y. Kuramoto and T. Yamada, Turbulent state in chemical reactions, ibid. 56 (1976), 679.
• [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer, New York, 1972.
• [18] D. Michelson, Steady solutions of the Kuramoto-Sivashinsky equation, Phys. D 19 (1986), 89-111.
• [19] D. Michelson, Bunsen flames as steady solutions of the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 23 (1991), 364-386.
• [20] D. Michelson, Stability of the Bunsen flame profiles in the Kuramoto-Sivashinsky equation, SIAM J. Math. Anal. 27 (1996), 765-781.
• [21] L. Molinet, Local dissipativity in L₂ for the Kuramoto-Sivashinsky equation in spatial dimension two, J. Dynam. Differential Equations, to appear.
• [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.
• [23] B. Nicolaenko and B. Scheurer, Remarks on the Kuramoto-Sivashinsky equation, Phys. D 12 (1984), 391-395.
• [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.
• [25] F. W. J. Olver, Introduction to Asymptotics and Special Functions, Acad. Press, New York, 1974.
• [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.
• [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.
• [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.
• [29] G. I. Sivashinsky, On flame propagation under conditions of stoichiometry, SIAM J. Appl. Math. 39 (1980), 67-82.
• [30] G. I. Sivashinsky, Instabilities, pattern formation, and turbulence in flames, Ann. Rev. Fluid Mech. 15 (1983), 179-199.
• [31] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1958.
• [32] G. Tolstov, Fourier Series, Dover, New York, 1962.
• [33] J. Topper and T. Kawahara, Approximate equations for nonlinear waves on a viscous fluid, J. Phys. Soc. Japan 44 (1978), 663-666.
• [34] W. C. Troy, The existence of steady solutions of the Kuramoto-Sivashinsky equation, J. Differential Equations 82 (1989), 269-313.
• [35] V. V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), 619-634.
• [36] V. V. Varlamov, On spatially periodic solutions of the damped Boussinesq equation, ibid. 10 (1997), 1197-1211.
• [37] V. V. Varlamov, On the initial-boundary value problem for the damped Boussinesq equation, Discrete Contin. Dynam. Systems 4 (1998), 431-444.
• [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] V. V. Varlamov, The third-order nonlinear evolution equation governing wave propagation in relaxing media, Stud. Appl. Math. 99 (1997), 25-47.
• [40] V. V. Varlamov, On the damped Boussinesq equation in a circle, Nonlinear Anal. 38 (1999), 447-470.
• [41] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, London, 1966.