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Global Steady Prandtl Expansion over a Moving Boundary I

Peking Mathematical Journal volume 2, pages 155–238 (2019)Cite this article

Abstract

This is the first of three papers in which we prove that steady, incompressible Navier–Stokes flows posed over the moving boundary, \(y = 0\), can be decomposed into Euler and Prandtl flows in the inviscid limit globally in \([1, \infty ) \times [0,\infty )\), assuming a sufficiently small velocity mismatch. In this part, sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers.

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Notes

  1. It is clear that by rescaling \(z \rightarrow (1-\delta )^{\frac{1}{2}}z\), we can replace the factor of \(1-\delta\) in front of \(\psi ''\) by simply 1. This rescaling would change the main linear operator, \((1-\delta ) \psi '' + \frac{z}{2} \psi '\) to \(\psi '' + \frac{z}{2} \psi '\). For notational ease, then, we work simply with the \(\psi ''\) instead of \((1-\delta )\psi ''\). The actual self-similar variable, then, is really \((1-\delta )\frac{\eta }{\sqrt{x}}\), but as \((1-\delta )\) is near 1, this causes no confusion in the analysis to follow.

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Acknowledgements

The author thanks Yan Guo for many valuable discussions regarding this research. The author also thanks Bjorn Sandstede for introducing him to the paper [5].

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  1. Fine Hall, Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Sameer Iyer

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  1. Sameer Iyer
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Correspondence to Sameer Iyer.

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This research was completed under partial support by NSF Grant 1209437.

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Iyer, S. Global Steady Prandtl Expansion over a Moving Boundary I. Peking Math J 2, 155–238 (2019). https://doi.org/10.1007/s42543-019-00011-4

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