Abstract. 



It is well-known that the pressure of a lubricating fluid

filling the gap between two solid surfaces satisfies the Reynolds equation involving the distance function, h, between both planes, as a crucial coefficient. Nevertheless, in most of the applications the function h is not known a priori. Here we consider the simple case in which the

surfaces are two parallel planes and assume prescribed the total force applied upon one of the surfaces. We give some sufficient conditions on the total force in order to solve this inverse problem. We show that in the incompressible case, such a condition is also necessary

Díaz Díaz, Jesús Ildefonso

Tello del Castillo, José Ignacio

Abstract. 



It is well-known that the pressure of a lubricating fluid filling the gap between two solid surfaces satisfies the Reynolds equation involving the distance function, h, between both planes, as a crucial coefficient. Nevertheless, in most of the applications the function h is not known a priori. Here we consider the simple case in which the surfaces are two parallel planes and assume prescribed the total force applied upon one of the surfaces. We give some sufficient conditions on the total force in order to solve this inverse problem. We show that in the incompressible case, such a condition is also necessary

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A note on some inverse problems arising in lubrication theory

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A NOTE ON SOME INVERSE PROBLEMS ARISING IN 
LUBRICATION THEORY 
J . I . DÍAZ AND J . I . TELLO 
Abs t r ac t . It is well-known that the pressure of a lubricating fluid 
filling the gap between two solid surfaces satisfies the Reynolds equation 
involving the distance function, h, between both planes, as a crucial 
coefñcient. Nevertheless, in most of the applications the function h is 
not known a priori. Here we consider the simple case in which the 
surfaces are two parallel planes and assume prescribed the total forcé 
applied upon one of the surfaces. We give some sufficient conditions on 
the total forcé in order to solve this inverse problem. We show that in 
the incompressible case, such a condition is also necessary. 
1. INTRODUCTION 
Since the pioneering work by O. Reynolds, in 1886, it has been well-
known that the pressure of a lubricating fluid filling the gap between two 
solid surfaces satisfies the so-called Reynolds equation involving the distance 
function, h, between both planes, as a crucial coefíicient. Nevertheless, in 
most of the applications the function h is not known a priori. The hard disc 
of computers or the compact disc player are two examples of the many real 
situations where this kind of problem appears. 
Although several works have been devoted to the study of this problem 
when some extra information is added to the formulation (see, e.g., the ar-
ticles Bayada [1], and Bayada and El Alaoui Talibi [2], in which the total 
load supported by the surfaces is prescribed), the necessity of imposing suit-
able conditions on the additional information in order to get a well-posed 
formulation seems not well observed. 
Here we consider the simple case in which the surfaces are two parallel 
planes, and so the unknown distance between both planes is merely a time 
function h = h(t), for t G (0, T) with T > 0 given. So, the unknowns are the 
functions (h(t),P(x,y,t)), where P denotes the pressure and (x,y) G Í2, an 
open and bounded set of R2. We assume given the initial distance between 
the planes 
h(0) = ho, (1.1) 
the external pressure Pa (a positive constant), the initial pressure distribu-
tion P(x,0) = Po(x) (only for the case of a compressible fluid), and the 
relative velocity (U, V) of the superior plañe (in fact here assumed to be a 
constant vector). In this note we also assume to be known the total forcé 
applied upon the superior plañe and that it has only a nonzero component, 
F(t), in the z-direction (orthogonal to the planes). 
The main goal of the note is to give some sufficient conditions on F(t) in 
order to solve this inverse problem. Moreover, in the incompressible case, 
we shall show that our sufficient condition on F(t) is also necessary for the 
existence of a solution (h, P). We recall that in the case of an incompressible 
fluid, under the above conditions, the Reynolds equation deals with the linear 
elliptic inverse problem: assuming F(t) is known, find (h, P) such that 
-div(h(tfVP) = -h'(t), in Q x (0, T), 
P = Pa, ondÜx(0,T) (1.2) 
F(t) = Jn(P(x,y,t) - Pa)dxdy for í e (0,T). 
In spite of the simplicity of the above formulation, it seems that the study of 
necessary and sufficient conditions on F(t) was not clearly indicated before 
in the literature. 
The compressible case is more delicate since the associated problem be-
comes parabolic and of quasilinear type. To simplify the formulation, we 
consider the simpler case in which the spatial domain is reduced to a one-
dimensional interval / = (0, L) (so, there is no dependence of data and 
unknowns with respect to the y variable). Then, under some conditions on 
the degree of compressibility of the gas (see, e.g., Friedman and Tello [4]) we 
arrive at the following inverse problem for the Reynolds equation: assuming 
F(t) is known, find (h, P) such that 
(
^ ^ + Uh(t)^-el((ah(tf + mtfP)^) = 0, in / x (0,T), 
P(x,0) = P0(x), i n / , 
P(0,t) = P(L,t) = Pa, 
F(t)= I(P(x,y,t)-Pa)dxdy, f o r í e ( 0 , T ) , 
Jn 
(1.3) 
where Pa, a, (3, e, and U are known positive constants and T is small enough. 
2. INCOMPRESSIBLE CASE 
Problem (1.1) and (1.2) can be solved by using the auxiliary problem 
-Aw = 1, in Q, 
w = 0, on díl. (2.1) 
Consider 
K(ü) = / w(x,y)dxdy, 
Jn 
and assume that 
F(t)> PaKf\ f€(0 ,oo) . (2.2) 
Theorem 1. 
a) / / 
F(s)ds>-1^- V í e ( 0 , o o ) , (2.3) 
then there exists a unique solution (h(t),P(x,y,t)) of the problem 
(1.1), (1.2) such that 
1. 
PTíT = ~ M ^and theref°re> sicJn{h') = -sign(F(t))). (2.4) 
Jn particular, 
"
(t)
 =
 l 4 + i fMÍ F ( s ) ' í s r i 
2. 
^ , y , í ) = | ^ ( ^ y ) + í V (2.5) 
b) 7/ í/iere exists to > 0 such that 
to
 K(Q) /"* 7ÍÍQ) 
F(s)ds = W- and / F(s)ds < ^ V í e ( 0 , í o ) , (2.6) 
o 2/i0 y0 2/i0 
then h(t) —> oo w/ien t y to and P(x,y,to) = ximw{xiy) + -Pa-
problem (2.1), and h is defined as the unique solution of the problem 
Proof. We construct P = -WJ^W + Pa, where w is the solution of the 
that is, 
Vi G (0, oo j ; 
1 2 "' 
By construction of P we know that 
r n , . , min{F(í),0} , , . . ^ 
min {P(x,y,t)} > —— max{w(x,y)} + Pa. (x,y)eU K{il) (x,y)eU 
From (2.2) we obtain that P > 0; furthermore, 
{P{x,y,t)-Pa)dxdy= / ——w(x,y)dxdy = F(t). 
By uniqueness of the problem (2.1), the solution P(x,y,t) of (1.1) is 
h'(t) 
P{x, y, t) = -—-w{x, y) + P0-
Substituting in (1.1) we obtain that (P,h) is the solution of the problem 
(1.1), (1.3). If F(t) satisfies (2.3), we obtain (2.4), (2.5), and h exists there 
for all t G (0, oo). 
If F(t) satisfies (2.6), then 
and solving the O.D.E. for h(t) we obtain that 
Kt) = [ p + ^ y J F(s)ds]-* Vi G (0, í0), 
and therefore 
F(t) 
P(x,y,t) = YJfñw + pa,Vt G (0,í0). 
By taking limits when t / ¿o, we obtain that h(to) = +oo, which proves 
that the sufficient condition (2.6) is also a necessary assumption. Notice 
that, even if h(to) = +oo, the function P is well defined at this time ¿o since 
P(x,y,tQ) = ^ w + Pa. 
We recall that the derivation of the Reynolds equation requires knowing 
that the function gap h is small enough. So, the above estimates allow 
us to get some quantitative conditions in terms of F(t) to justify such a 
derivation. We also notice that the uniqueness of the couple (h(t),P(x, y, t)) 
is a consequence of the uniqueness of the equation h'(t) = —jao)h3(t)-
Remark 2.1. By (2.1) we know | |AIÍ;| |¿2(^) < \Q\i. Taking w as a test 
function in (2.1), we obtain ||io||#i(Q) = (K(Q,))^, and deduce that 
IMIff2 (n )nHi (n ) < M* +(K(SÍ))%. 
Since H2(ü) -^> L°°(Q) is a continuous inclusión, it results that ||HIL°°(Í3) < 
C(Q), and therefore 
P>mm{0,F(í)}^H+Pa. 
If min{F(í)} > -PaK(Ü)/C(Ü), we obtain that P > 0. 
3. COMPRESSIBLE CASE 
In this section we study the one-dimensional compressible problem (1.3) 
and (1.1) for small time T(Pa, PQ, ho, F, F') (see (3.13)). The following exis-
tence result requires again a total forcé F(t) small enough. 
Theorem 2. Let P0 > 0 such that P0 - Pa e W¿'°°(0, L), and let F G 
C1(0,T), satisfying 
L Po(s)ds ,
 N , 
h
°
 =
 F(0)+PaL' ° < F° < F(t) + PaL < Fh P o > 0 , (3.1) 
for some positives constants FQ < F\. Then, there exists e* > 0 such 
that if 0 < e < e* problem (1.3) and (1.1) has, at least, one solution 
(h(t),P(x,y,t)). 
Proof. Limit case: e = 0. In the special case e = 0, the system (1.3), (1.1) 
reduces to 
^ + í / ^ = 0, 0<x<L, 0<t<T, 
P{x, 0) = PQ{X), 0<x<L, (3.2) 
P(0,t)=Pa, 0<t<T. 
The solution "Ph" of (3.2) is given by Ph = <p = <p(x — Ut), where 
,n(*\-í hopo(s), ifO <s<L, 
^
s ) - \ h{-fj)Pa, if - L < S < 0 . 
Integrating, we obtain 
cL rL 
and then 
F(t)= ¡ {P{x,t)-Pa)dx = J {^{x *Jt) - Pa)dx, 
! rL 
*
( í )
 = F(t) + PaL l Vi*-™)** 
1 fut 1 fL 
=
 F / , x , v T(pa h(t - — x)dx + h0 / P0(x-Ut)dx). 
£ (t) + faL 7 0 17 7{/ t 
Introducing the change of variables s = t — jjx, r = x — Ut, we obtain 
i rt r-L-Ut 
Ht)=
 PaL(PaUJ h(s)ds + hoj P0{r)dr). (3.3) 
By (3.1) we obtain that the unique solution h of (3.3) is positive, and by 
defining 
P(X t) rtx~Ut) 
it results that (h(t),P(x,t)) is the unique solution of problem (3.2). 
Remark 3.1. Notice that h G C^O, §). 
Case e > 0. Let £ := hP. Then 
^ + U^--e—(ah^-+ph^^-)=0, 0 < a; < L, 0 < í < T. (3.4) 
ot ox ox ox ox 
Through this section we use the constants 
ao := -ho > 0, a\ := 2ho + 1 < oo, (3.5) 
M := (h0+l)(Pa+ max {P0(x)}), m := ^ mm{Pa, min {P0(x)}}, (3.6) 
2 A/f 7" F 1 
F2 := max |F'(í)|, a2 :=— -~-^ - + — < oo. (3.7) 
te[o,£] F¿ Pa 
We consider the closed, convex set G C C°(0,T), defined by 
G = { / Í 6 t71,oo(0, T), fc(0) = /¿o, a0 < h(t) < a i < oo, |/V(í)| < a2}. 
We introduce the truncature function <fi defined by 
s, H ao < s < c¿i, 
(s) = { a0, if s < a0, 
a\, if a\ < s. 
It is useful to start considering the truncated problem 
-é + U^-e—{ah^ + f3h^) = ^ 0<x<L, 0<t<T, (3.8) 
ai ox ox ox ox 
Z(0,t)=Z(L,t) = Pah(t), 0<t<T, (3.9) 
£(x, 0) = h0P0(x) > 0, 0 < a; < L, (3.10) 
^( í ) + PaL = h. (3.11) 
Proposition 3.1. Given h <E G and £o > 0 the problem (3.8)-(3.10) has a 
unique weak solution satisfying m < £ < M. 
Proof. We consider the problem 
i + ^ S - £ ¿ ^ + f ^ ) ) = ° ' o<x<L>o<t<T 
£(0,t)=Z(L,t) = Pah(t), 0<t<T, 
£(x, 0) = ho(P0(x) -Pa)>0, 0<x<L. 
Since (a£ + §|£|£) is an increasing function of £ and h > ao > 0, it results 
that the operator 
Qii d d 3 
A(
^
 :=Udx~~edx:(hdx~(au + 2^U + Pa^u + Pa^ 
is a maximal monotone operator in Hl(0, L), and then there exists a unique 
weak solution £ e L2(0,T : H\0,L)) nC([0,T] : L2(0,L)) with § e 
L2(0,T : (ff^O.L))'). Taking - (£ - m)~ and (£ - M)+ as test function 
in (3.8) we obtain 
dt ( / [{Í-m)-]
2dx+ / [(£ - M)+]2dx) < 0. 
Wo Jo ' 
Since m < £(x,0) < M, we deduce m < £(x,t) < M; then |£| = £, and we 
get that £ satisfies (3.8). D 
Lemma 3.1. Let h G G and £ 6e í/ie solution of the problem (3.8)-(3.11), 
where 
. r M¿F2[7 ^ , 
£ m m l F o ( a a i f / + 2/3aiM(Paa2 + l ) ) ' a i /3 (P a a 2 + l ) t " ' 
(3.12) 
Then 
I — I < ——— , at the points x = 0,L. 
ox u 
Proof. Let £ = Pah + Xx, where A = f V g + 1 . We obtain 
<9t 9a; e—iah^f- + ¡3hí 
_ Pg«2   
9{, 
._
 v-.„_ , , ^ _ ,=Patí + U\-eh/3\2. 
ox ox ox 
By the choice of A and since e < e*, it results that Pah' + [7 A — eh[3\2 > 0. 
Since { > { in a; = 0, L it results that £ is a supersolution of the problem 
(3.8)-(3.11). In the same way we prove that £ = Pah(t) + A(l — x) is a 
supersolution in a neighborhood of L. Taking 
í = Pah a2 in x G (0, 7V¿(í)f7^ í e ( 0 : L, (7 «2 (7 
we obtain that £ is a subsolution and £ = ^ + 77 (^ ~~ -0 1S a^ so a subsolution 
in a neighborhood of L. By comparison it results that 
«2
 = 9 £ < 9 e < 9 e = x 
U dx ~ dx ~ dx 
and we also deduce that 
a2
 dí 
U dx dx dx 
which proves the lemma. 
We consider the function T : G 
r(h) = <x Jo £(x,t)dx. 
on x = 0, 
on x = L, 
i defined by 
= h, 
D 
F(t) + PaL 
where £ is the solution of the problem (3.8)-(3.10); then we have 
Lemma 3.2. T has a fixed point in G. 
Proof. By construction of <fi we know ao < h(t) < a\, and furthermore, 
integrating by parts in (3.8) we get 
/ £tdx = e(ah + /%£) — 
7o °x 
As a consequence of Lemma 3.1, we obtain that 
{tda; 
From (3.11) it results that 
L 
dh 
< e(aai + 2f3aiM 7V*2_KL 
U ' 
,,(lo í(x^)dx\ \I0L^m^dx ¡0LÍ{x,t)dxF'{t) 
V F(t) + PaLJl F(t) + PaL (F(t) + PaL)< 
and we deduce that 
| - | < - ( a a x + 2 / t a x M — ^ ) + - ^ - . 
By (3.12) and (3.7) it results that |^ | | < a2, and we deduce that JmT C G. 
Since the inclusión W1,QO(0, j,) C C°(0, ^ ) is continuous and compact, JmT 
is a compact subset of C°. Applying Schauder's fixed-point theorem we 
obtain the wished-for result. D 
End of the proof of Theorem 2. It is clear that each fixed point of T is 
a solution of the problem (3.8)-(3.11). We denote by (£e,/&e) the solution of 
the problem (3.8)-(3.11). Since \h'^\ < a2, for all 0 < e < e*, we obtain that 
h0 - a2T < he{t) <h0 + a2T. Then, by taking 
T<min{^,§}, (3.3, 
we get that c¿o <he{t) < a i , for all 0 < e < e*, and the refere 
h - AÍ i'o ^x^)dx\ _ Jo£Á%,t)dx 
lie F(t) + PaL J F(t) + PaL • 
Then (£
€
,h
€
) is a solution of the problem (3.8)-(3.11), and (P
€
 = j^,he) 
satisfies (1.2), (1.1) which conclude the proof. D 
Acknowledgment. The authors thank G. Bayada for some useful conver-
sations. 
REFERENCES 
[1] G. Bayada, "Inequations Variationelles Elliptiques avec Conditions aux Limites Peri-
odiques: Application a la Resolution de PEquation de Reynolds," These de 3 e m e eyele, 
U. of Lyon, 1972. 
[2] G. Bayada and M. El Alaoui Talibi, Control by the coefficients in a variational in-
equality: the inverse elastohydrodynamic lubrication problem, Nonlinear Analysis: Real 
World Applications, 1 (2000), 315-328. 
[3] A. Friedman and B. Hu, Head-media interaction in magnetic recording, Archive Ratio-
nal Mech. Anal., 140 (1997), 79-101. 
[4] A. Friedman and J.I. Tello, Head-media interaction in magnetic recording, J. Diff. 
Equations, 171 (2001), 443-461. 
[5] J.I. Tello, "Tratamiento Matemático de Procesos Físicos con Difusión y Convección," 
Ph.D. Thesis, U. Complutense de Madrid, 2001. 


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