Let Ω be a bounded open set in R 2 with “smooth” boundary ∂Ω. Ω is the region where two solid bodies are in contact. These two solids are moving and the average of their velocities is denoted by V=(V 1 ,V 2 ). The pressure p=p(X) which develops in a fluid layer confined between these two bodies satisfies the so- called Reynolds lubrication equation:

(1)∇·(h 3 ρ∇p)=6μV·∇(ρh)inΩ

(2)p=p a on∂Ω·

Here ρ is the density of the fluid, μ>0 its dynamic viscosity, h=h(X) is the distance between the two bodies and p a ≥0 is the given ambient pressure.

We will assume all along that h is a Lipschitz continuous function such that

(3)0<h 1 ≤h(X)≤h 2 a·e·X=(x,y)∈Ω

(4)|∇h(X)|≤Ha·e·X=(x,y)∈Ω

where h 1 , h 2 , H are positive constants. When the fluid is incompressible, i.e. ρ is a positive constant, existence and uniqueness for a solution to (1), (2) is a trivial matter since the equation is linear. In the compressible case when (5) ρ=ρ(p) the problem becomes more challenging.

The paper is organized as follows: in Section 1 we investigate the problem of existence of a solution for (1), (2) when ρ is given by (5). In Section 2 we consider a case where (1), (2) reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution. Finally, in a third section we give some extensions of the results of Section 2 in higher dimension.

Chipot, M (1988). *On the Reynolds lubrication equation.* Nonlinear Analysis: Theory, Methods & Applications, 12(7):699-718.

## Abstract

Let Ω be a bounded open set in R 2 with “smooth” boundary ∂Ω. Ω is the region where two solid bodies are in contact. These two solids are moving and the average of their velocities is denoted by V=(V 1 ,V 2 ). The pressure p=p(X) which develops in a fluid layer confined between these two bodies satisfies the so- called Reynolds lubrication equation:

(1)∇·(h 3 ρ∇p)=6μV·∇(ρh)inΩ

(2)p=p a on∂Ω·

Here ρ is the density of the fluid, μ>0 its dynamic viscosity, h=h(X) is the distance between the two bodies and p a ≥0 is the given ambient pressure.

We will assume all along that h is a Lipschitz continuous function such that

(3)0<h 1 ≤h(X)≤h 2 a·e·X=(x,y)∈Ω

(4)|∇h(X)|≤Ha·e·X=(x,y)∈Ω

where h 1 , h 2 , H are positive constants. When the fluid is incompressible, i.e. ρ is a positive constant, existence and uniqueness for a solution to (1), (2) is a trivial matter since the equation is linear. In the compressible case when (5) ρ=ρ(p) the problem becomes more challenging.

The paper is organized as follows: in Section 1 we investigate the problem of existence of a solution for (1), (2) when ρ is given by (5). In Section 2 we consider a case where (1), (2) reduces to a one dimensional problem and we study the question of uniqueness as well as the shape of the solution. Finally, in a third section we give some extensions of the results of Section 2 in higher dimension.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Nonlinear elliptic equations; existence; uniqueness; smoothness |

Language: | English |

Date: | 1988 |

Deposited On: | 28 Oct 2009 13:11 |

Last Modified: | 05 Apr 2016 13:29 |

Publisher: | Elsevier |

ISSN: | 0362-546X |

Publisher DOI: | https://doi.org/10.1016/0362-546X(88)90023-5 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=947883 |

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