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Some results on the compressible Reynolds equation. (English) Zbl 0697.76085

Contributions to nonlinear partial differential equations, Vol. II, Proc. 2nd Franco-Span. Colloq., Paris 1985, Pitman Res. Notes Math. Ser. 155, 79-90 (1987).
[For the entire collection see Zbl 0614.00011.]
Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ 2\) with a smooth boundary \(\Gamma\) : \(\Omega\) is the region of the plane where two solid bodies are in close contact. These two bodies are moving and the sum 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 solids satisfies the so-called Reynolds lubrication equation: \(\nabla \cdot (h^ 3\rho\) \(\nabla p)=6\mu V\cdot \nabla(\rho h)\) in \(\Omega\); \(p=p_ a\) on \(\Gamma\). Here \(\rho\) is the density of the fluid, \(\mu >0\) its dynamic viscosity, \(h=h(X)\) is the distance between the two bodies and \(p_ a\geq 0\) is the given ambient pressure (we refer to the author and M. Luskin, IMA Vol. Math. Appl. 3, 61-75 (1987; Zbl 0656.76062) for a derivation of this equation using a simplified version of the Navier-Stokes equations).
We assume that \(h\) is a Lipschitz continuous function such that: \(0<h_ 1\leq h(X)\leq h_ 2\) a.e. \(X=(x,y)\in \Omega\); \(| \nabla h(X)| \leq H\) a.e. \(X=(x,y)\in \Omega\), where \(h_ 1,h_ 2\), \(H\) are positive constants.
Moreover, we assume that the fluid is compressible that is to say that \(\rho\) depends on \(p\) and so we have: \[ (*)\quad \nabla \cdot (h^ 3\rho (p)\nabla p)=6\mu \quad V\cdot \nabla (\rho (p)h)\text{ in } \Omega;\quad p=p_ a \text{ on }\Gamma. \] When the fluid obeys the isothermal perfect gas relation we have (after rescaling) (**) \(\rho(p)=P.\)
We present here some new results for (*) which go beyond the scope of (**). Our assumption on \(\rho\) will be \(\rho\) is continuous, non decreasing, \(\rho(0)=0\) and \(\rho(s)>0\) \(\forall s>0\). This is reasonable for a density. Note that some of our techniques apply also to more general models.
First we recall how to get existence of a solution to (*). Then we show some techniques leading to uniqueness and in the last section we do a rough study of the shape of the solution. For complete proofs and details the reader is referred to another work by the author [Nonlinear Anal., Theory Methods Appl. 12, No.7, 699-718 (1988; Zbl 0668.76073)].

MSC:

76N15 Gas dynamics (general theory)