A rigorous derivation of the time-dependent Reynolds equation (Q2856525)

The authors study the incompressible Stokes equation for the fluid velocity \(u\) and the pressure \(P\). They consider a boundary value problem in a 3-d thin domain, when the fluid domain is bounded by two moving surfaces, and study the asymptotic behavior of the solution as the distance of the two boundary surfaces approaches to zero. At first they solve the boundary value problem when the distance of these two surfaces is of order \(\varepsilon >0\). Introducing the notion of a weak solution to this case, they prove the existence and the uniquness of the weak solution \(u^\varepsilon\). Letting \(\varepsilon\rightarrow +0\), one obtains a weak limit \(u^\ast\), and a limit pressure \(P^\ast\). It is shown that these limit functions satisfy the Reynolds equation. In this way, they give a rigorous discussion for this physical phenomenon from mathematical point of view.