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Asymptotics in Fluid Mechanics

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Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations

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Abstract

As we pointed out in some detail in Chapter 1, the concept of first arose in two physical contexts: propagation in a certain spatial region of a gene, whose carriers have an advantage in the struggle for existence, and fluid mechanics, in particular, and involving shocks. In Chapters 3 and 4, we studied the asymptotic behaviour of solutions of nonlinear PDEs of parabolic type, which include those describing gene propagation. In the present chapter, we discuss some physical problems which arise from fluid mechanics and which are governed by hyperbolic or parabolic systems of equations. These systems admit similarity solutions of the first or second kind. The latter, in general, enjoy intermediate asymptotic character in some parametric regimes. We recall that the self-similar solutions of the first kind are fully determined by the dimensional considerations and require the solution of the resulting nonlinear ODEs with appropriate conditions at the shock and the centre of the in this context, say, whereas those of the second kind involve solution of an eigenvalue problem in the reduced phase plane, the , for example.

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Authors and Affiliations

  1. Department of Mathmetics, Indian Institute of Technology Madras, Sardar Patel Road, 600036, Chennai, India

    Srinivasa Rao Ch.

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  1. Sachdev* P.L.
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  2. Srinivasa Rao Ch.
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Correspondence to Srinivasa Rao Ch. .

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P.L., S., Ch., S. (2009). Asymptotics in Fluid Mechanics. In: Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87809-6_5

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