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

Durany, J.; García, G.; Vázquez, C.

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)

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

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