Durany, J.; García, G.; Vázquez, C. A mixed Dirichlet-Neumann problem for a nonlinear Reynolds equation in elastohydrodynamic piezoviscous lubrication. (English) Zbl 0857.35044 Proc. Edinb. Math. Soc., II. Ser. 39, No. 1, 151-162 (1996). 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. Reviewer: V.Shelukhin (Novosibirsk) Cited in 5 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 76D08 Lubrication theory 35R35 Free boundary problems for PDEs Keywords:Schauder’s fixed point theorem; nonnegative weak solution PDF BibTeX XML Cite \textit{J. Durany} et al., Proc. Edinb. Math. Soc., II. Ser. 39, No. 1, 151--162 (1996; Zbl 0857.35044) Full Text: DOI References: [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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.