Stability and bifurcation in a modulated Burgers system

Authors:
W. E. Olmstead and S. H. Davis

Journal:
Quart. Appl. Math. **39** (1982), 467-477

MSC:
Primary 35B32; Secondary 35Q20, 58E07, 76E99

DOI:
https://doi.org/10.1090/qam/644101

MathSciNet review:
644101

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate . Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in . For each of these oscillatory-type modulations, it is found that has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

**[1]**J. M. Burgers,*Mathematical example illustrating relations occurring in the theory of turbulent fluid motion*, Trans. Roy. Neth. Acad. Sci. Amsterdam**17**, 1-53 (1939)**[2]**J. M. Burgers,*A mathematical model illustrating the theory of turbulence*, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., 1948, pp. 171–199. edited by Richard von Mises and Theodore von Kármán,. MR**0027195****[3]**G. B. Whitham,*Linear and nonlinear waves*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR**0483954****[4]**Wiktor Eckhaus,*Studies in non-linear stability theory*, Springer Tracts in Natural Philosophy, Vol. 6, Springer-Verlag New York, New York, Inc., 1965. MR**0196298****[5]**Carmine Golia and Jacob M. Abel,*Path integral synthesis of Lyapunov functionals for partial differential equations*, Internat. J. Non-Linear Mech.**10**(1975), no. 6, 333–345 (English, with French and German summaries). MR**0430489****[6]**C. O. Horgan and W. E. Olmstead,*Stability and uniqueness for a turbulence model of Burgers*, Quart. Appl. Math.**36**(1978/79), no. 2, 121–127. MR**0495602**, https://doi.org/10.1090/S0033-569X-1978-0495602-6**[7]**S. H. Davis,*The stability of time-periodic flows*, Ann. Rev. Fluid Mech.**8**, 57-74 (1976)**[8]**S. H. Davis,*Finite amplitude instability of time-dependent flows*, J. Fluid Mech.**45**, 33-48 (1971)**[9]**R. C. DiPrima and J. T. Stuart,*Nonlocal effects in the stability of flow between eccentric rotating cylinders*, J. Fluid Mech.**54**, 393-416 (1972)**[10]**P. Hall,*The stability of unsteady cylinder flows*, J. Fluid Mech.**67**, 29-64 (1975)**[11]**L. A. Rubenfeld,*A model bifurcation problem exhibiting the effects of slow passage through critical*, SIAM J. Appl. Math.**37**(1979), no. 2, 302–306. MR**543949**, https://doi.org/10.1137/0137021**[12]**B. J. Matkowsky,*A simple nonlinear dynamic stability problem*, Bull. Amer. Math. Soc.**76**(1970), 620–625. MR**0257544**, https://doi.org/10.1090/S0002-9904-1970-12461-2**[13]**W. C. Reynolds and M. C. Potter,*Finite-amplitude instability of parallel shear flows*, J. Fluid Mech.**27**, 465-492 (1967)**[14]**A. Davey,*The growth of Taylor vortices in flow between rotating cylinders*, J. Fluid Mech.**14**(1962), 336–368. MR**0145788**, https://doi.org/10.1017/S0022112062001287**[15]**L. A. Segel and J. T. Stuart,*On the question of the preferred mode in cellular thermal convection*, J. Fluid Mech.**13**(1962), 289–306. MR**0140237**, https://doi.org/10.1017/S0022112062000683

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
35B32,
35Q20,
58E07,
76E99

Retrieve articles in all journals with MSC: 35B32, 35Q20, 58E07, 76E99

Additional Information

DOI:
https://doi.org/10.1090/qam/644101

Article copyright:
© Copyright 1982
American Mathematical Society