Stability and bifurcation in a modulated Burgers system

Olmstead, W. E.; Davis, S. H.

doi:10.1090/qam/644101

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
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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 $A\left ( \tau \right )$ of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate $\theta \left ( \tau \right )$. Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in $\theta \left ( \tau \right )$. For each of these oscillatory-type modulations, it is found that ${A^2}\left ( \tau \right )$ 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.

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