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The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n, with application to plane poiseuille flow

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Abstract

The existence of periodic solutions of the Navier-Stokes equations in function spaces based upon (L p(Ω))nis proved. The paper has three parts, (a) A proof of the existence of strong solutions of the evolution equation with initial data in a solenoidal subspace of (L p(Ω))n. (b) The evolution equation is restricted to a space of time periodic functions and a Fredholm integral equation on this space is formed. The Lyapunov-Schmidt method is applied to prove the existence of bifurcating time periodic solutions in the presence of symmetry. (c) The theory is applied to the bifurcation of periodic solutions from planar Poiseuille flow in the presence of symmetry (SO(2) x O(2) x S 1) yielding new results for this classic problem. The O(2) invariance is in the spanwise direction. With the periodicity in time and in the streamwise direction we find that generically there is a bifurcation to both oblique travelling waves and to travelling waves that are stationary in the spanwise direction. There are however points of degeneracy on the neutral surface. A numerical method is used to identify these points and an analysis in the neighborhood of the degenerate points yields more complex periodic solutions as well as branches of quasi-periodic solutions.

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References

  1. T. S. Chen & D. D. Joseph [1973] Subcritical Bifurcation of Plane Poiseuille Flow, J. Fluid Mech. 58, pp. 337–351.

    Google Scholar 

  2. M. G. Crandall & P. H. Rabinowitz [1978] The Hopf Bifurcation Theorem in Infinite Dimensions, Arch. Rational Mech. Anal. 67, pp. 53–72.

    Google Scholar 

  3. P. G. Drazin & W. Reid [1981] Hydrodynamic Stability, Cambridge Univ. Press.

  4. T. Erneux & B. J. Matkowsky [1984] Quasi-Periodic Waves Along a Pulsating Propagating Front in a Reaction Diffusion Equation, SIAM J. Appl. Math. 44, pp. 536–544.

    Google Scholar 

  5. A. Friedman [1976] Partial Differential Equations, Krieger Publishing.

  6. H. Fujita & T. Kato [1964] On the Navier-Stokes Initial Value Problem, Arch. Rational Mech. Anal. 16, pp. 269–315.

    Google Scholar 

  7. D. Fujiwara & H. Morimoto [1977] An L p-Theorem of the Helmholtz Decomposition of Vector Fields, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 24, pp. 685–700.

    Google Scholar 

  8. A. Georgescu [1985] Hydrodynamic Stability Theory, Martinus Nijhoff Publishers, Romania.

    Google Scholar 

  9. Y. Giga [1981] Analyticity of the Semi-group Generated by the Stokes Operator in L p Spaces, Math. Z., 178, pp. 297–329.

    Google Scholar 

  10. Y. Giga [1985] Domains of Fractional Powers of the Stokes Operator in L p Spaces, Arch. Rational Mech. Anal. 89, pp. 251–265.

    Google Scholar 

  11. Y. Giga & T. Miyakawa [1985] Solutions in L p Of the Navier-Stokes Initial Value Problem, Arch. Rational Mech. Anal. 89, pp. 267–281.

    Google Scholar 

  12. M. Golubitsky & M. Roberts [1987] A Classification of Degenerate Hopf Bifurcation with O(2) Symmetry, J. Diff. Eqns. 69, pp. 216–264.

    Google Scholar 

  13. M. Golubitsky & D. Schaeffer [1985] Singularity and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, New York.

    Google Scholar 

  14. M. Golubitsky & I. Stewart [1985] Hopf Bifurcation in the Presence of Symmetry, Arch. Rational Mech. Anal. 87, pp. 107–165.

    Google Scholar 

  15. D. Gottlieb & S. A. Orszag [1977] Numerical Analysis of Spectral Methods: Theory and Applications, SIAM Monograph CBMS-NSF Series.

  16. T. Healey [1988] Global Bifurcation and Continuation in the Presence of Symmetry with an Application to Solid Mechanics, SIAM J. Math. Anal. 19, pp. 824–840.

    Google Scholar 

  17. M. Heard & S. Rankin [1987] Weak Solutions of a Parabolic Integrodifferential Equation, to appear in J. Math. Anal. Appl.

  18. T. Herbert [1976] Periodic Secondary Motions in a Plane Channel, Proc. 5th Conf. on Num. Fl. Dyn., LNP # 59, Springer-Verlag, pp. 235–240.

  19. G. Iooss [1972] Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d'évolution du type Navier-Stokes, Arch. Rational Mech. Anal. 47, pp. 301–329.

    Google Scholar 

  20. G. Iooss [1984] Bifurcation and Transition to Turbulence in Hydrodynamics, in Bifurcation Theory and Applications, LNM # 1057, Springer-Verlag, pp. 152–201.

  21. V. I. Iudovich [1971] The onset of Auto-oscillations in a fluid, J. Appl. Math. Mech. 35, pp. 587–603.

    Google Scholar 

  22. J. Ize [1979] Periodic Solutions of Nonlinear Parabolic Equations, Comm. PDE's, 4 (12), pp. 1299–1387.

    Google Scholar 

  23. D. D. Joseph [1976] Stability of Fluid Motions I, Springer-Verlag.

  24. D. Joseph & D. Sattinger [1972] Bifurcating Time Periodic Solutions and Their Stability, Arch. Rational Mech. Anal. 45, pp. 79–109.

    Google Scholar 

  25. T. Kato [1984] Perturbation Theory for Linear Operators, Springer-Verlag.

  26. H. Kielhöfer [1970] Hopf Bifurcation at Multiple Eigenvalues, Arch. Rational Mech. Anal. 69, pp. 53–83.

    Google Scholar 

  27. E. Knobloch [1986] On the Degenerate Hopf Bifurcation with O(2) Symmetry, in Contemporary Mathematics, Vol.. 56 (Golubitsky & Guckenheimer, Eds.), AMS, pp. 193–201.

  28. E. Knobloch, A. E. Deane, J. Toomre, & D. R. Moore [1986] Doubly Diffusive Waves, in Contemporary Mathematics, Vol. 56 (Golubitsky and Guckenheimer, Eds.), AMS, pp. 203–216.

  29. T. Miyakaya [1981] On the Initial Value Problem for the Navier-Stokes Equations in L p Spaces, Hiroshima Math. J. 11, pp. 9–20.

    Google Scholar 

  30. S. A. Orszag [1971] Accurate Solution of the Orr-Sommerfeld Stability Equation, J. Fluid Mech. 50, pp. 689–703.

    Google Scholar 

  31. J. Pugh [1988] Finite Amplitude Waves in Plane Poiseuille Flow, Ph. D. Thesis in Applied Math., Cal Tech.

  32. S. Rankin [1987] Semilinear Evolution Equations in Banach Spaces with Applications to Parabolic PDE's, preprint.

  33. W. Rudin [1973] Functional Analysis, McGraw-Hill.

  34. P. G. Saffman [1983] Vortices, Stability, and Turbulence, in Annals of the N.Y. Academic of Science 404, pp. 12–24.

    Google Scholar 

  35. D. Sattinger [1971] Bifurcation of Periodic Solutions of the Navier-Stokes Equations, Arch. Rational Mech. Anal. 41, pp. 66–80.

    Google Scholar 

  36. D. H. Sattinger [1983] Branching in the Presence of Symmetry, SIAM Monograph CMBS-NSF Series #40.

  37. J. Serrin [1959] On the Stability of Viscous Fluid Motions, Arch. Rational Mech. Anal. 3, pp. 1–13.

    Google Scholar 

  38. A. E. Taylor & D. C. Lay [1980] Introduction to Functional Analysis, 2nd Edition, John Wiley.

  39. A. Vanderbauwhede [1978] Alternative Problems and Symmetry, J. Math. Anal. Appl. 62, pp. 483–494.

    Google Scholar 

  40. A. Vanderbauwhede [1982] Local Bifurcation and Symmetry, Pitman, Res. Notes in Math. #75.

  41. F. Weissler [1981] The Navier-Stokes Initial Value Problem in L p, Arch. Rational Mech. Anal. 74, pp. 219–230.

    Google Scholar 

  42. J. Zahn, J. Toomre, E. Spiegel, & D. Gough [1974] Nonlinear Cellular Motions in Poiseuille Channel Flow, J. Fluid Mech. 64, pp. 319–345.

    Google Scholar 

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  1. Department of Mathematics, Worcester Polytechnic Institute, Worcester, Massachusetts

    Thomas J. Bridges

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  1. Thomas J. Bridges
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Communicated by M. Golubitsky

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Bridges, T.J. The Hopf bifurcation with symmetry for the Navier-Stokes equations in (Lp(Ω)) n, with application to plane poiseuille flow. Arch. Rational Mech. Anal. 106, 335–376 (1989). https://doi.org/10.1007/BF00281352

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