Parabolic Nonlinear Second Order Slip Reynolds Equation: Approximation and Existence

Small scale gaseous lubrication theory is used widely in mic roelectromechanical systems (MEMS) such as microbearings, micropumps and microturbine s. B cause of microsize or even nanosize geometries flow can no longer be considered as a continuum. This failure of the Navier-Stokes description as the characteristic sys tem scale approaches the mean free path makes this problem both challenging and interesti ng from a scientific point of view. The deviation from Navier-Stokes is quantified by the Knudse n numberKn [1, 2] typically defined as the ratio of the molecular mean free path to e characteristic system scale. It is well known that flow can be classified into three ca tegories [3]:Kn ≤ 10 the flow can be considered as a continuum; Kn > 10 the flow is considered to be a free molecular flow;10 ≤ Kn ≤ 10 the flow can neither be a continuum flow nor a free molecular one. The conventional Navier-Stokes syste m is based on a continuum assumption and it is no longer valid if the Knudsen number is b eyond a certain limit [4]. This involves the selection of an appropriate model and boun dary conditions. The well-known Reynolds equation in the continuum regime is [5,6]:


Introduction
Small scale gaseous lubrication theory is used widely in microelectromechanical systems (MEMS) such as microbearings, micropumps and microturbines.Because of microsize or even nanosize geometries flow can no longer be considered as a continuum.This failure of the Navier-Stokes description as the characteristic system scale approaches the mean free path makes this problem both challenging and interesting from a scientific point of view.
The deviation from Navier-Stokes is quantified by the Knudsen number K n [1,2] typically defined as the ratio of the molecular mean free path to the characteristic system scale.It is well known that flow can be classified into three categories [3]: K n ≤ 10 −3 the flow can be considered as a continuum; K n > 10 the flow is considered to be a free molecular flow; 10 −3 ≤ K n ≤ 10 the flow can neither be a continuum flow nor a free molecular one.The conventional Navier-Stokes system is based on a continuum assumption and it is no longer valid if the Knudsen number is beyond a certain limit [4].This involves the selection of an appropriate model and boundary conditions.
The well-known Reynolds equation in the continuum regime is [5,6]: where h is the local gas bearing thickness, p the local pressure, ρ the local gas density, µ the viscosity, U 0 is the moving plate velocity and x = (x 1 , x 2 ) ∈ Ω ⊆ R 2 (with smooth boundary ∂Ω).
In the slip regime the above equation needs modifications.Taking the Hsia's second order model, the boundary condition are given as follows [7]: , U x2 are the velocity distributions, τ is the surface accommodation coefficient and λ is the mean free path.For these boundary conditions, the velocity distributions are obtained by solving the momentum equation [7]: The second order modified Reynolds equation can hence be obtained by incorporating the expressions of U x1 and U x2 into the continuity equation and then integrating from Normally, the non-dimensional second order slip Reynolds equation is used which is given by [6]: with Λ is the bearing vector.
In earlier works [8,9], existence and uniqueness for the stationary version of (1) was proved under some hypotheses on the data.In this paper, we continue our investigation concerning the same problem and we plan to establish existence of weak solutions to (1) through a procedure of semi-discrete scheme as in [10] and to present a priori estimates for this scheme.First, in Section 2, we present the problem with corresponding boundary and initial data and we introduce a new formulation, Section 3 is devoted to some notations and to semi-discrete scheme.A priori estimates for this scheme are established in Section 4 and existence of weak solution is proved in the last section.

Formulation of the problem
Here, we consider the following problem where Ψ > 0 is the ambient pressure and T a positive number.We assume that the functions h : Ω×]0, T [→ R satisfies the following hypotheses: We introduce the new unknown function and we consider the function g : ]0, +∞[→ R such that It is easy to see that g is an increasing and bijective function.We have from the above equality with Our initial problem (P) becomes in u By the weak solution of problem (P u ), we mean a function u such that: we will prove existence of a positive solution p for the problem (P) without any knowledge on the sign of u.

A semi-discrete time scheme
We replace in (P u ) the time derivative by the backward difference quotient.Let N be a positive integer and τ = T N , denote by {U j } j=0,1,..,N the solution of the elliptic system: The variational formulation associated to problem (P u ) τ is given by: This stationary problem is very close to the one studied in [8], so we can use the same arguments as in [8] to prove existence and uniqueness, under some hypotheses on the data (6), of solution for the problem P u ) τ V at each time step.
Theorem 1.For j = 1, 2, . . ., N and under the hypothesis (where C p is the constant of Poincaré [11] and Λ e is the Euclidean norm of Λ), there exists a weak solution U j ∈ H 1 (Ω) to the problem (P u ) τ V .In addition, We have uniqueness among all weak solutions to problem (P u ) τ V .Further, suppose that U j i is a weak solution to (P u ) τ V corresponding to the boundary data
For the following, C denotes a generic constant which can take different values in different occurrences and we consider that ( 6) is verified for j = 1, 2, .., N and that U 0 (x) = u 0 (x) ≥ 0.
Proposition 1.For a big enough value of K n , there exists a constant C which don't rely on τ such that Proof.By choosing v = (U j − Ψ j u )τ in (P u ) τ V and summing on j, we get which leads, according to (A 1 ), to However, from the definition of χ τ and the fact that we obtain Using the Cauchy-Schwarz inequality, we get The last term on the right hand side of ( 7) can be estimated as: which can be written J = J 1 + J 2 such that Due to the monotonicity of the function s → κ(., ., s), we have that since the term Ω U 0 0 κ(x, 0, s) dsdx is nonnegative, we have which is nonnegative, due to the monotonicity of κ.Now, using the fact that κ is Lipschitz with respect to the second term and estimate (8), we get on the other hand it follows that The term J 2 is treated as follows Finally, by substituting the above estimates into (7) and using embedding inequalities, we obtain which achieves the proof of the proposition.Proposition 2. There exists a constant C independent on τ such that V and summing on j from 1 to n, we get χ(x, jτ, U j ) ∇κ(x, jτ, U j ) − ∇κ(x, jτ, Ψ j u ) dx = 0, Now, we estimate successively the last three terms of (10).
With the help of the Cauchy-Schwarz inequality we get .
From the definition of κ, we have κ(., jτ, and from the previous proposition, we gain On the other hand, using the Cauchy-Schwarz inequality in the elliptic term of (10) gives The same calculation for the convective term of (10) leads to However, from ( 8), (9) and Proposition 1, it is easy to see that by substituting the above estimates into (10), we obtain using the previous proposition which give the desired estimate.
Here, we give a strong convergence of κ(x, t, U τ ) in L 1 (Q).
Next, we will prove that κ = ∂κ(.,.,u)Finally, we can pass to the limit in (14) and we find that u satisfy (5).Remark 3. Particular domain Ω (for which C p is small enough) is necessary to get assumptions of Theorem 2 while keeping K n ≤ 10.