## Abstract

This paper is concerned with the stationary problem of an aero-taxis system with physical boundary conditions proposed by Tuval et al (2005 Proc. Natl Acad. Sci. 102 2277-82) to describe the boundary layer formation in the air-fluid interface in any dimensions. By considering a special case where fluid is free, the stationary problem is essentially reduced to a singularly perturbed nonlocal semi-linear elliptic problem. Denoting the diffusion rate of oxygen by ϵ > 0, we show that the stationary problem admits a unique classical solution of boundary-layer profile as ϵ → 0, where the boundary-layer thickness is of order ϵ. When the domain is a ball, we find a refined asymptotic boundary layer profile up to the first-order approximation of ϵ by which we find that the slope of the layer profile in the immediate vicinity of the boundary decreases with respect to (w.r.t.) the curvature while the boundary-layer thickness increases w.r.t. the curvature.

Original language | English |
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Pages (from-to) | 5111-5141 |

Number of pages | 31 |

Journal | Nonlinearity |

Volume | 33 |

Issue number | 10 |

DOIs | |

Publication status | Published - 13 Aug 2020 |

## Keywords

- boundary layer
- chemotaxis
- nonlocal
- semi-linear elliptic equation

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics