A mixed Dirichlet-Neumann problem for a nonlinear Reynolds equation in elastohydrodynamic piezoviscous lubrication. (English) Zbl 0857.35044

Given a rectangular \(\Omega=\{(x_1,x_2):|x_i|<1\}\), the first goal is to find a nonnegative weak solution \(p\in W^{1,2}(\Omega)\) of the equation \[ \text{div}(h^3 [p]\exp(-\alpha p)\nabla p)={\partial\over{\partial x_1}}c(H_\varepsilon(p) h[p]) \] with the boundary conditions \(p=0\) at \(\{|x_2|=1\}\cup\{x_1=1\}\) and \({\partial\over{\partial x_1}}p=c(H_\varepsilon(p)- \theta_0)\) at \(\{x_1=-1\}\). Here \(c\) and \(\alpha\) are positive constants, \(h[p]=h_0(x)+ p*1/|x|\), \(h_0\) is a smooth positive function, \(H_\varepsilon(p)\), \(\varepsilon\to 0\), is an approximation of the Heaviside function. Then the estimates \(|p|_{L^\infty}\leq c\) and \(|p|_{W^{1,2}}\leq c\) are obtained uniformly in \(\varepsilon\) to prove the convergence \(p\to\overline{p}\) to a weak solution \(\overline{p}\) of the limit equation with \(H_\varepsilon(p)\) substituted by \(\theta(x,t)=w \text{-lim }H_\varepsilon(p)\). It is claimed that the pair \(\overline{p}\) and \(\theta\) solves the lubricant pressure problem modelled by the Reynolds equation.


35J65 Nonlinear boundary value problems for linear elliptic equations
76D08 Lubrication theory
35R35 Free boundary problems for PDEs
Full Text: DOI


[1] Bayada, Existence of Solution for Elastohydrodynamic Piezoviscous Lubrication Problems with a New Model for Cavitation 163 (1994)
[2] Brezis, C. R. Acad. Sci. Paris Sér pp 711– (1978)
[3] Bayada, J. Theor. Appl. Mech. 5 pp 703– (1986)
[4] Bayada, Boll. Un. Mat. Ital. 6 pp 543– (1984)
[5] Vázquez, Adv. in Math. Sci. and Appl. 4 pp 313– (1994)
[6] Rodrígues, European J. Appl. Math. 4 pp 83– (1993)
[7] DOI: 10.1016/0020-7225(85)90075-8 · doi:10.1016/0020-7225(85)90075-8
[8] Kinderlehrer, An Introduction to Variational Inequalities and their Applications (1980) · Zbl 0457.35001
[9] DOI: 10.1137/0521002 · Zbl 0718.35101 · doi:10.1137/0521002
[10] DOI: 10.1080/00036819408840250 · Zbl 0841.35133 · doi:10.1080/00036819408840250
[11] Gilbarg, Elliptic Partial Differential Equations of Second Order (1977) · doi:10.1007/978-3-642-96379-7
[12] Dowson, Elastohydrodynamic Lubrication (1977)
[13] Chipot, Variational Inequalities and Flow on Porous Media 52 (1984) · doi:10.1007/978-1-4612-1120-4
[14] Cameron, Basic Lubrication Theory (1981)
[15] DOI: 10.1002/mma.1670180402 · Zbl 0820.35110 · doi:10.1002/mma.1670180402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.