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Global Steady Prandtl Expansion over a Moving Boundary I

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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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