Justification of the linear long-wave approximation to viscous fluid flow down an inclined plane

Authors:
S. M. Sun and M. C. Shen

Journal:
Quart. Appl. Math. **52** (1994), 759-775

MSC:
Primary 76D05; Secondary 35Q30, 76D33

DOI:
https://doi.org/10.1090/qam/1306049

MathSciNet review:
MR1306049

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Abstract: The objective of this paper is to justify the linear long-wave approximation used in the derivation of approximate equations for long waves on the free surface of a two-dimensional viscous fluid flow down an inclined plane. To the first order of a small parameter, the approximate equation is a heat equation, which becomes ill-posed if a Reynolds number is greater than some critical value . To overcome this difficulty we consider a higher-order approximate equation, which is well-posed even if , and show that the solution of the higher-order equation is an approximation to the solution of the linearized Navier-Stokes equations. The justification is based upon a set of long-wave initial conditions, and the error bounds can also be expressed in terms of pointwise estimates.

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DOI:
https://doi.org/10.1090/qam/1306049

Article copyright:
© Copyright 1994
American Mathematical Society