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January  2015, 2(1): 95-142. doi: 10.3934/jcd.2015.2.95

Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof

Piotr Zgliczyński 1,

1. 

Institute of Computer Science, Jagiellonian University, Lojasiewicza 6, 30-348 Kraków, Poland

Received  May 2012 Revised  January 2015 Published  August 2015

We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.
Keywords: dissipative PDEs., Bifurcations, rigorous numerics.
Mathematics Subject Classification: Primary: 35B32, 37G10, 65G2.
Citation: Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Rational Mech. An., 197 (2010), 1033-1051. doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

, CAPD - Computer assisted proofs in dynamics, a package for rigorous numerics,, Available from: , ().   Google Scholar

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[4]

S.-N. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.  Google Scholar

[5]

F. Christiansen, P. Cvitanovic and V. Putkaradze, Spatiotemporal chaos in terms of unstable recurrent patterns, Nonlinearity, 10 (1997), 55-70. doi: 10.1088/0951-7715/10/1/004.  Google Scholar

[6]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D, 67 (1993), 321-326. doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[7]

E. J. Doedel, AUTO: a program for the bifurcation analysis of autonomous system, Congr. Numer., 30 (1981), 265-284.  Google Scholar

[8]

E. J. Doedel and R. C. Paffenroth, The AUTO2000: command line user interface, in Proceedings of the 9-th Python Conference, 2001, 233-241. Google Scholar

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C. Foias, B. Nicolaenko, G. 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.  Google Scholar

[10]

J. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation; A bridge between PDEs and dynamical systems, Physica D, 18 (1986), 113-126. doi: 10.1016/0167-2789(86)90166-1.  Google Scholar

[11]

J. S. Il'yashenko, Global Analysis of the Phase Portrait for the Kuramoto-Sivashinsky equation, J. Dyn. Diff. Eq., 4 (1992), 585-615. doi: 10.1007/BF01048261.  Google Scholar

[12]

M. Jolly, I. Kevrekidis and E. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D, 44 (1990), 38-60. doi: 10.1016/0167-2789(90)90046-R.  Google Scholar

[13]

M. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky Equation, Adv. Differential Equations, 5 (2000), 31-66.  Google Scholar

[14]

M. Jolly, R. Rosa and R. Temam, Acurate computations on inertial manifolds, SIAM J. Sci. Compt., 22 (2000), 2216-2238. doi: 10.1137/S1064827599351738.  Google Scholar

[15]

I. Kevrekidis, B. Nicolaenko and C. Scovel, Back in saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), 760-790. doi: 10.1137/0150045.  Google Scholar

[16]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.  Google Scholar

[17]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426. doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[18]

R. E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, N.J., 1966.  Google Scholar

[19]

A. Neumeier, Interval Methods for Systems of Equations, Cambrigde University Press, 1990.  Google Scholar

[20]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, 37. Springer-Verlag, New York, 2000.  Google Scholar

[21]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames - 1. Derivation of basic equations, Acta Astron, 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[22]

C. Scovel, I. Kevrekidis and B. Nicolaenko, Scaling laws and the prediction of bifurcations in systems modeling pattern formation, Physics Letters A, 130 (1988), 73-80. doi: 10.1016/0375-9601(88)90242-3.  Google Scholar

[23]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288. doi: 10.1007/s002080010010.  Google Scholar

[24]

P. Zgliczyński, Trapping regions and an ODE-type proof of existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Univ. Iag. Acta Math., 41 (2003), 89-113.  Google Scholar

[25]

P. Zgliczyński, On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach, J. Diff. Eq., 195 (2003), 271-283. doi: 10.1016/j.jde.2003.07.009.  Google Scholar

[26]

P. Zgliczyński, Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof, SIAM Journal on Applied Dynamical Systems, 1 (2002), 215-235. doi: 10.1137/S111111110240176X.  Google Scholar

[27]

, the file containing numerical data from the bifurcation proofs,, Available from: , ().   Google Scholar

show all references

References:
[1]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Rational Mech. An., 197 (2010), 1033-1051. doi: 10.1007/s00205-010-0309-7.  Google Scholar

[2]

, CAPD - Computer assisted proofs in dynamics, a package for rigorous numerics,, Available from: , ().   Google Scholar

[3]

L. Cesari, Functional analysis and Galerkin's method, Mich. Math. Jour., 11 (1964), 385-414. doi: 10.1307/mmj/1028999194.  Google Scholar

[4]

S.-N. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.  Google Scholar

[5]

F. Christiansen, P. Cvitanovic and V. Putkaradze, Spatiotemporal chaos in terms of unstable recurrent patterns, Nonlinearity, 10 (1997), 55-70. doi: 10.1088/0951-7715/10/1/004.  Google Scholar

[6]

P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D, 67 (1993), 321-326. doi: 10.1016/0167-2789(93)90168-Z.  Google Scholar

[7]

E. J. Doedel, AUTO: a program for the bifurcation analysis of autonomous system, Congr. Numer., 30 (1981), 265-284.  Google Scholar

[8]

E. J. Doedel and R. C. Paffenroth, The AUTO2000: command line user interface, in Proceedings of the 9-th Python Conference, 2001, 233-241. Google Scholar

[9]

C. Foias, B. Nicolaenko, G. 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.  Google Scholar

[10]

J. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation; A bridge between PDEs and dynamical systems, Physica D, 18 (1986), 113-126. doi: 10.1016/0167-2789(86)90166-1.  Google Scholar

[11]

J. S. Il'yashenko, Global Analysis of the Phase Portrait for the Kuramoto-Sivashinsky equation, J. Dyn. Diff. Eq., 4 (1992), 585-615. doi: 10.1007/BF01048261.  Google Scholar

[12]

M. Jolly, I. Kevrekidis and E. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D, 44 (1990), 38-60. doi: 10.1016/0167-2789(90)90046-R.  Google Scholar

[13]

M. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky Equation, Adv. Differential Equations, 5 (2000), 31-66.  Google Scholar

[14]

M. Jolly, R. Rosa and R. Temam, Acurate computations on inertial manifolds, SIAM J. Sci. Compt., 22 (2000), 2216-2238. doi: 10.1137/S1064827599351738.  Google Scholar

[15]

I. Kevrekidis, B. Nicolaenko and C. Scovel, Back in saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), 760-790. doi: 10.1137/0150045.  Google Scholar

[16]

Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369. doi: 10.1143/PTP.55.356.  Google Scholar

[17]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426. doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[18]

R. E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, N.J., 1966.  Google Scholar

[19]

A. Neumeier, Interval Methods for Systems of Equations, Cambrigde University Press, 1990.  Google Scholar

[20]

A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, 37. Springer-Verlag, New York, 2000.  Google Scholar

[21]

G. I. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames - 1. Derivation of basic equations, Acta Astron, 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0.  Google Scholar

[22]

C. Scovel, I. Kevrekidis and B. Nicolaenko, Scaling laws and the prediction of bifurcations in systems modeling pattern formation, Physics Letters A, 130 (1988), 73-80. doi: 10.1016/0375-9601(88)90242-3.  Google Scholar

[23]

P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288. doi: 10.1007/s002080010010.  Google Scholar

[24]

P. Zgliczyński, Trapping regions and an ODE-type proof of existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Univ. Iag. Acta Math., 41 (2003), 89-113.  Google Scholar

[25]

P. Zgliczyński, On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach, J. Diff. Eq., 195 (2003), 271-283. doi: 10.1016/j.jde.2003.07.009.  Google Scholar

[26]

P. Zgliczyński, Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof, SIAM Journal on Applied Dynamical Systems, 1 (2002), 215-235. doi: 10.1137/S111111110240176X.  Google Scholar

[27]

, the file containing numerical data from the bifurcation proofs,, Available from: , ().   Google Scholar

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