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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.

MSC:

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

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