Compressible flows: New existence results and justification of the Reynolds asymptotic in thin films

About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication

“…They show that the JFO model can be obtained as the limit of this compressible model and propose some numerical procedures to solve it. Although being somewhat heuristic, this result has been supported by recent rigorous mathematic results [14] for the asymptotic thin film compressible Navier Stokes system. The interest of the more physical compressible Elrod-Adams model is that there is only one unknown (the density of lubricant) instead of two (P and θ) in the JFO model.…”

“…η in which z 1 η and z 2 η are two solutions for (14) and γ > 0 is a real given parameter. Multiplying the difference of the two equations satisfied by z 1 η and z 2 η , by the test function z + γ+z + which belongs to H 1 0 (Ω), we get…”

Reynolds' Limit Formula for Dorodnitzyn's Atmospheric Boundary Layer Model in Convective Conditions

“…Let ρ, u, v, T, p, κ, µ be as in Definition 3. Suppose they satisfy the Dorodnitzyn's Boundary Layer Model given by equations (1), (2), (3), (4), (5), (7), (8) with boundary conditions (12), (13), (14), (15), (16), (17), p(x, y) = p (x, h (x)) ∀(x, y) ∈ Ω h , and ∂u/∂x = 0 a.e. in Ω h .…”

A rigorous derivation of the stationary compressible Reynolds equation via the Navier–Stokes equations

“…Although the issue of compactness of solutions to problem (1.1) is nowadays relatively well understood, see the seminal monograph by Lions [12] as well as other numerous recent extensions of the theory listed in Plotnikov and Weigant [19], the problem of suitable a priori bounds in the case on the non-homogeneous boundary conditions and a proper construction of solutions seems largely open. Chupin and Sard [5] applied the framework proposed by Bresch and Desjardins [6], where the viscosity coefficients depend on the density in a specific way. This approach requires additional friction term in the momentum equation, div(̺u ⊗ u) + ∇p(̺) = divS(∇u) − r̺|u|u, (1.5) as well an extra boundary condition for the density,…”