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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
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