Boundary Value Problems
Volume 2009, Article ID 708389,15pages doi:10.1155/2009/708389
Research Article
Existence of Weak Solutions for
a Nonlinear Elliptic System
Ming Fang
1and Robert P. Gilbert
21Department of Mathematics, Norfolk State University, Norfolk, VA 23504, USA
2Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
Correspondence should be addressed to Ming Fang,[email protected]
Received 3 April 2009; Accepted 31 July 2009
Recommended by Kanishka Perera
We investigate the existence of weak solutions to the following Dirichlet boundary value problem, which occurs when modeling an injection molding process with a partial slip condition on the boundary. We have−Δθkθ|∇p|rqxinΩ;−div{kθ|∇p|r−2βx|∇p|r0−2∇p}0 inΩ;
θθ0, andpp0on∂Ω.
Copyrightq2009 M. Fang and R. P. Gilbert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Injection molding is a manufacturing process for producing parts from both thermoplastic and thermosetting plastic materials. When the material is in contact with the mold wall surface, one has three choices:ino slipwhich implies that the material sticks to the surface
iipartial slip, andiiicomplete slip1–5 . Navier6 in 1827 first proposed a partial slip condition for rough surfaces, relating the tangential velocityvαto the local tangential shear
stressτα3
vα−βτα3, 1.1
where β indicates the amount of slip. Whenβ 0, 1.1 reduces to the no-slip boundary condition. A nonzeroβimplies partial slip. Asβ → ∞, the solid surface tends to full slip.
Problem 1. Find functionsθandpdefined inΩsuch that
−Δθkθ∇prqx inΩ, 1.2
−divkθ∇pr−2βx∇pr0−2∇p0 inΩ, 1.3
θθ0, pp0 on∂Ω. 1.4
Here we assume thatΩis a bounded domain inRN with aC1 boundary. We assume also thatq,θ0,p0,β, andkare given functions, whiler is a given positive constant related to
the power law index;pis the pressure of the flow, andθis the temperature. The leading order termβx|∇p|r0−2of the PDE1.3is derived from a nonlinear slip condition of Navier type.
Similar derivations based on the Navier slip condition occur elsewhere, for example,8,9 ,
10, equation2.4 .
The mathematical model for this system was established in7 . Some related papers, both rigorous and formal, are3,11–13 . In11,13 , existence results in no-slip surface,β0, are obtained, while in 3, 7 , Navier’s slip conditions, β /0 and r0 0, are investigated,
and numerical, existence, uniqueness, and regularity results are given. Although the physical models are two dimensional, we shall carry out our proofs in the case of N
dimension.
InSection 2, we introduce some notations and lemmas needed in later sections. In
Section 3, we investigate the existence, uniqueness, stability, and continuity of solution p
to the nonlinear equation 1.3. In Section 4, we study the existence of weak solutions to Problem1.
Using Rothe’s method of time discretization and an existence result for Problem1, one can establish existence of week solutions to the followingtime-dependentproblem.
Problem 2. Find functionsθandpdefined inΩT such that
θt−Δθkθ∇prqx inΩT,
−divkθ∇pr−2βx∇pr0−2∇p0 inΩ
T,
θθ0, pp0 on∂Ω×0, T,
θϕ onΩ× {0}.
1.5
2. Notations and Preliminaries
2.1. Notations
In this paper, fors > 1,letH1,sΩandH1,s
0 Ωdenote the usual Sobolev space equipped
with the standard norm. Let
σ
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
N
N−1, if 1< r < N,
r
r−1, if r > N,
qN
qN−qN, if rN,
2.1
whereN < q <∞. The conjugate exponent ofσis
σ∗
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
N, if 1< r < N,
r, ifr > N,
qN
q−N, ifr N.
2.2
We assume that the boundary values θ0 and p0 for Problem 1 can be extended to
functions defined onΩsuch that
θ0∈H1,σΩ, p0 ∈H1,τΩ. 2.3
We further assume that there exist positive numbersk2> k1>0 andβ0such that
k1< kθ< k2, ∀θ∈R1,
0≤βx≤β0.
2.4
Finally, we assume that forθm, θ∈H01,σΩ θ0, limm→ ∞θmθa.e. inΩindicates
lim
m→ ∞kθm kθ a.e. inΩ. 2.5
For the convenience of exposition, we assume that
1< r0< r < τ <∞. 2.6
2.2. Preliminaries
An important inequalitye.g., see11, page 550 in the study ofp-Laplacian is as follows:
|x|r−2x−yr−2yx−y≥
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ax−yr, ifr≥2,
ax−y2
b|x|y2−r, if 1< r <2,
2.7
wherea >0 andb >0 are certain constants.
To establish coercivity condition, we will use the following inequality:
abr ≤2rarbr, 2.8
wherer >0,a >0, andb >0.
Using the Sobolev Embedding Theorem and H ¨older’s Inequality, we can derive the following resultsfor more details, see11, Lemma 3.4 and13, Lemma 4.2 .
Lemma 2.1. The following statements hold
iFor any positive numbersαandς, ifu∈LαΩandv∈LςΩ,then
uv∈Lγ, whereγ
1
α
1
ς
−1
; 2.9
moreover, uvLγΩ≤uLαΩvLςΩ.
iiIfp∈H1,rΩand1< r < N, thenp|∇p|r−2∇p∈LN/N−1Ω N;moreover,
p∇pr−2∇p
LN/N−1Ω≤pLNr/N−rΩ∇prL−r1Ω. 2.10
iiiIfp∈H1,rΩand1< r <∞, then|∇p|r−2∇p∇p
0∈LζΩ, where
ζ
1
r∗
1
τ
−1
, 2.11
andr∗denotes the conjugate of r, namely,r∗r/r−1for1< r <∞;moreover,
∇pr−2∇p∇p0
LζΩ≤∇p
r−1
LrΩ∇p0LτΩ. 2.12
ivIfp∈H1,rΩandn≤r <∞, then
p∇pr−2∇p∈Lr∗Ωn, r > n,
p∇pr−2∇p∈LsΩ n, rn,
wheres 1/r∗1/q−1andr < q <∞. Moreover
p∇pr−2∇p
Lr∗Ω≤C∇p
r−1
LrΩ, r > n,
p∇pr−2∇p
LsΩ≤pLqΩ∇p
r−1
LrΩ, rn.
2.14
The existence proof will use the following general result of monotone operators14, Corollary III.1.8, page 87 and15, Proposition 17.2 .
Proposition 2.2. LetK⊂Xbe a closed convex set (/φ), and letΛ:K → Xbe monotone, coercive, and weakly continuous onK. Then there exists
u∈K:Λu, v−u ≥0 for anyv∈K. 2.15
The uniqueness proof is based on a supersolution argumentsimilar definition can be found in15, Chapter 3 .
Definition 2.3. A functionu∈Hloc1,rΩis a weak supersolution of the equation
−divkθ|∇u|r−2βx|∇u|r0−2∇u0 2.16
inΩif
Ω
kθ|∇u|r−2βx|∇u|r0−2∇u· ∇ϕ dx≥0, 2.17
wheneverϕ∈C0∞Ωis nonnegative.
3. A Dirichlet Boundary Value Problem
We study the following Dirichlet boundary value problem:
−divkθ∇pr−2βx∇pr0−2∇p0 inΩ,
pp0 on∂Ω.
3.1
Definition 3.1. We say thatpθ−p0∈H01,rΩis a weak solution to3.1if
Ω
kθ∇pθr−2βx∇pθr0−2
∇pθ· ∇ξ dx0 3.2
Theorem 3.2. Assume that conditions 2.1–2.6 are satisfied. Then there exists a unique weak solutionpθto the Dirichlet boundary value problem3.1in the sense ofDefinition 3.1. In addition, the solutionpθsatisfies the following properties.
1we have
pθH1,rΩ≤C, 3.3
whereCis a constant independent ofθandpθ; 2iflimm→ ∞θmθa.e. inΩ, then
lim
m→ ∞pθm pθ strongly inH
1,rΩ.
3.4
The idea behind the existence proof is related to15,16 . We will first consider the following Obstacle Problem.
Problem 3. Find a functionpinKψ,p0such that
Ω
kθ∇pr−2βx∇pr0−2∇p∇ξ−pdx≥0 3.5
for allξ∈Kψ,p0. Here
Kψ,p0Ω
p∈H1,rΩ:p≥ψ a.e. inΩ, p−p0∈H01,rΩ
. 3.6
Lemma 3.3. IfKψ,p0is nonempty, then there is a unique solution p to the Problem 3 inKψ,p0.
Proof ofLemma 3.3.Our proof will useProposition 2.2. LetXLrΩ;Rnand write
K∇v:v∈Kψ,p0
. 3.7
It follows from the proof in15, Proposition 17.2 thatK⊂Xis a closed convex set. Next we define a mappingΛ:K → Xby
Λv, u
Ω
kθ|v|r−2βx|v|r0−2vu dx ∀u∈X. 3.8
By H ¨older’s inequality,
|Λv, u| ≤k2vLr−rΩ1 uLrΩβ0vLr0r−01ΩuLr0Ω
≤CvrL−r1ΩvrL0r−01Ω
uLrΩ.
3.9
Here we used Assumption2.6, that is, 1 < r0 < r < τ < ∞. Therefore we haveΛv ∈ X
To show thatΛis coercive onK, fixϕ∈K. Then
Λu−Λϕ, u−ϕ
Ω
kθ|u|r−2β|u|r0−2
u−kθϕr−2βϕr0−2ϕu−ϕdx
Ωkθ
|u|r−2u−ϕr−2ϕu−ϕdx
Ωβ
|u|r0−2u−ϕr0−2
ϕu−ϕdx
≥
Ωkθ
|u|r−2u−ϕr−2ϕu−ϕdx
≥k1
urϕr−k2
ur−1ϕϕr−1u
≥k12−ru−ϕr−k22r−1ϕu−ϕr−1ϕr−1
−k2ϕr−1ϕ−uϕ. 3.10
Inequality2.8is used to arrive at the last step. This implies thatΛis coercive onK.
Finally, we show thatΛ is weakly continuous onK. Let ui ∈ K be a sequence that
converges to an elementu∈KinLrΩ. Select a subsequenceu
ij such thatuij → ua.e. inΩ.
Then it follows that
kθuij
r−2uijβ
uij
r0−2uij −→kθ|u|
r−2uβ|u|r0−2u 3.11
a.e. inΩ. Moreover,
Ω kθuij
r−2uijβ
uij
r0−2uij
r/r−1dx≤C
Ω
uij
ruij
r×r0−1/r−1 dx ≤C Ω uij
rdx
Ω uij
rdx
r0−1/r−1
≤C.
3.12
Thus we have that
kθuij
r−2uijβ
uij
r0−2uij kθ|u|
r−2uβ|u|r0−2u 3.13
weakly inLr/r−1Ω. Since the weak limit is independent of the choice of the subsequence, it follows that
kθ|ui|r−2uiβ|ui|r0−2ui kθ|u|r−2uβ|u|r0−2u 3.14
weakly inLr/r−1Ω. HenceΛis weakly continuous onK. We may applyProposition 2.2to
Our uniqueness proof is inspired by15, Lemmas 3.11, 3.22, and Theorem 3.21. Since
kθ|∇u|r−2βx|∇u|r0−2∇udoes not satisfy condition3.4ofAoperator in15 , we need
to prove the following lemma, which is equivalent to15, Lemma 3.11 . Then uniqueness can follow immediately from15, Lemma 3.22 .
Lemma 3.4. Ifu∈H1,rΩis a supersolution of 2.16inΩ, then
Ω
kθ|∇u|r−2βx|∇u|r0−2∇u· ∇ϕ dx≥0 3.15
for all nonnegativeϕ∈H01,rΩ.
Proof. Letϕ ∈ H01,rΩand choose nonnegative sequenceφi ∈ C∞0 Ωsuch thatϕi → ϕin
H1,rΩ. Equation2.6and H ¨older inequality imply that
Ω
kθ|∇u|r−2β|∇u|r0−2∇u· ∇ϕ dx−
Ω
kθ|∇u|r−2β|∇u|r0−2∇u· ∇ϕ
idx
Ωkθ|∇u|
r−2∇u· ∇ϕ−ϕ i
dx
Ωβ|∇u|
r0−2∇u· ∇ϕ−ϕ
i
dx
≤k2∇urL−rΩ1 ∇ϕ−ϕiLrΩβ0∇urL0r−01Ω∇ϕ−ϕiLr0Ω
≤C∇urL−r1Ωβ0∇urL0r−01Ω
∇ϕ−ϕiLrΩ.
3.16
Because limi→ ∞∇ϕ−ϕiLrΩ0, we obtain
Ω
kθ|∇u|r−2β|∇u|r0−2∇u· ∇ϕ dx lim
i→ ∞
Ω
kθ|∇u|r−2β|∇u|r0−2∇u· ∇ϕ
idx≥0 3.17
and the lemma follows.
Similar to15, Corollary 17.3, page 335 , one can also obtain the following Corollary.
Corollary 3.5. LetΩbe bounded andp0 ∈H1,rΩ. There is a weak solutionpθ ∈H01,rΩ p0to 3.1in the sense ofDefinition 3.1.
Proof ofTheorem 3.2. The existence result is given inCorollary 3.5, and we now turn to proof of uniqueness. For a givenθ, assume that there exists another solutionp1
θ.Then we have that
Δ:
Ω
k
θ∇pθr−2∇pθ−∇p1θ r−2
∇p1θ
β∇pθr0−2∇pθ−∇p1θ r0−2
∇pθ1
· ∇ξ dx0
for allξ ∈ H01,rΩ. If we takeξ pθ−p1θin above equation, from inequality2.7, we have
the following.
iwhenr≥2,
0 Δ
≥
Ωkθ
∇pθr−2∇pθ−∇p1θ r−2
∇p1θ
·∇pθ− ∇pθ1 dx ≥C Ω
∇pθ− ∇p1θ r
dx,
3.19
whereCis a positive constant;
iiwhen 1< r <2,
0 Δ
≥
Ωkθ
∇pθr−2∇pθ−∇pθ1 r−2
∇p1θ
·∇pθ− ∇p1θ dx ≥C Ω
∇pθ− ∇p1θ 2
b∇pθ∇pθ1 r−2
dx
≥C
Ω
∇p1θ− ∇pθ r
dx
2/r
Ω
b∇pθ∇p1θ r
dx
r−2/r
.
3.20
Here the H ¨older inequality for 0< t <1, namely,
Ωfgdx≥
Ω ftdx
1/t
Ω gt∗dx
1/t∗
, t∗ t
t−1 3.21
is applied to the last inequality.
Poincar´e’s inequality implies thatpθpθ1a.e. We complete the uniqueness proof.
Next we prove3.3. Takingξpθ−p0in3.2, we have
Ωkθ ∇pθr
dx≤
Ωkθ
∇pθr−2∇
pθ∇p0dx
Ωβ
∇pθr0−2∇
pθ∇p0dx. 3.22
From2.4, and the H ¨older inequality, we obtain
k1
Ω ∇pθr
dx≤k2
Ω ∇pθr
dx
r−1/r
Ω ∇p0r
dx
1/r
β0
Ω ∇pθr
dx
r0−1/r
Ω|∇p0|
r/r−r01dx
r−r01/r .
Young’s inequality withεimplies
k1
Ω ∇pθr
dx≤ε
Ω ∇pθr
dxC
Ω ∇p0r
dx
Ω
∇p0r/r−r01dx
3.24
and3.3follows immediately from2.3and2.6.
Finally, we prove3.4. From weak solution definition3.2, we know that
Ω
kθm∇pθm
r−2
β∇pθm
r0−2∇
pθm∇ξ dx
Ω
kθ∇pθr−2β∇pθr0−2
∇pθ∇ξ dx0.
3.25
Settingξpθm−pθand subtracting
Ωkθm|∇pθ|r−2β|∇pθ|r0−2∇pθ∇ξ dxfrom both sides,
we obtain that
Ω
kθm
∇pθm
r−2∇
pθm−∇pθ
r−2∇
pθ
β∇pθm
r0−2p
θm−∇pθ
r0−2∇p
θ
∇pθm −pθ
dx
Ωkθ−kθm
∇pθr−2∇pθ∇
pθm −pθ
dx.
3.26
Denote the right-hand side byΔ1. Similar to arguments in the uniqueness proof, we arrive at
the folloing:
iwhenr≥2,
C
Ω ∇pθ
m − ∇pθ
r
dx≤Δ1; 3.27
iiwhen 1< r <2,
C
Ω ∇pθ
m − ∇pθ
r
dx
2/r
Ω
b∇pθ∇pθm
r
dx
r−2/r
≤Δ1. 3.28
Egorov’s Theorem implies that for all >0, there is a closed subsetΩofΩsuch that|Ω\Ω|<
Integral implies
Δ1≤
Ω
Ω\Ω
|kθm−kθ|∇pθr−1∇
pθm−pθdx
≤ε
Ω
∇pθrdx r−1/r
2k2
Ω
∇pθm−pθ
r
dx
1/r
−→0 asθm−→θ.
3.29
Theorem 3.2is proved.
4. Nonlinear Elliptic Dirichlet System
Definition 4.1. We say that{θ, p}is a weak solution to Problem1if
θ−θ0∈H01,σΩ, p−p0∈H01,rΩ, 4.1
and for allv∈C0∞Ω
−
Ω∇θ· ∇v dx
Ω
kθ∇prqv dx, 4.2
and for allξ∈H01,rΩ
Ω
kθ∇pr−2βx∇pr0−2∇p· ∇ξ dx0.
4.3
Theorem 4.2. Assume that2.1–2.6hold. Then there exists a weak solution to Problem1in the sense ofDefinition 4.1.
We shall bound the critical growth,|∇p|r, on the right-hand side of4.2.
Lemma 4.3. Suppose thatθandpsatisfy
and4.3. Then, under the conditions ofTheorem 4.2, for allv∈C1Ω
Ωkθ ∇pr
vdx
Ωkθ
∇pr−2∇
p· ∇p0v dx
−
Ωkθ
∇pr−2∇
pp−p0
· ∇v dx
−
Ωβ
∇pr0−2∇p· ∇p−p
0
v dx
−
Ωβ
∇pr0−2∇pp−p
0
· ∇v dx.
4.5
Moreover, there exists a polynomialFthat is independent ofθandpsuch that
Ωkθ
∇prv dx≤FpH1,rΩ
vH1,σ∗Ω. 4.6
Proof. We first show4.5. Lettingξvp−p0in4.3, we obtain
Ωkθ
∇pr−2∇p·v∇p−p0
p−p0
∇vdx
Ωβ
∇pr−2∇p·v∇p−p0
p−p0
∇vdx0.
4.7
After some straightforward computations this yields exactly4.5.
We now show4.6. We denote the four terms on the right-hand side of equation4.5
by I, II, III, and IV, respectively. Under the conditions ofLemma 4.3, we have
∇pr−2∇
p∈Lr∗Ω, ∇p0∈LτΩ, r∗ r
r−1. 4.8
PartiiiofLemma 2.1and Sobolev’s imbedding theorems indicate
|I| ≤k2∇prL−rΩ1 ∇p0LτΩvLζ∗Ω
≤C∇prL−rΩ1 ∇p0LτΩvH1,σ∗Ω
≤C∇pLr−rΩ1 vH1,σ∗Ω,
4.9
whereζ∗τr/τ−rsatisfiesr−1/r1/τ1/ζ∗1.
According to Sobolev’s imbedding theorems, the integrability ofp−p0depends on
Case 11< r < N.
|II| ≤Cp−p0LNr/N−rΩ∇prL−r1Ω∇vLNΩ
≤CprH1,rΩprH−11,rΩ
vH1,NΩ.
4.10
Case 2rN.
|II| ≤Cp−p0LqΩ∇p
r−1
LrΩ∇vLqr/q−rΩ r < q <∞
≤CprH1,rΩprH−11,rΩ
vH1,qr/q−rΩ.
4.11
Case 3r > N. p−p0is a bounded continuous function, so
|II| ≤C∇prL−rΩ1 ∇vLrΩ. 4.12
We next estimate III:
|III| ≤
Ωβ ∇pr0
v dx
Ωβ
∇pr0−2∇
p· ∇p0v dx
≤β0∇prL0r−Ω1 vLr/r−r0ΩC∇p
r−1
LrΩvH1,σ∗Ω.
4.13
The estimate of the first term used H ¨older inequality and Sobolev’s imbedding theorems. The argument of the second estimate is similar to that of I.
Recall that 1< r0< r. Similar to II, we estimate IV in three different cases.
Case 11< r < N.
|IV| ≤Cp−p0LNr/N−rΩ∇prL0r−Ω1 ∇vLNr/Nr−r0rΩ. 4.14
SinceNr/Nr−r0 r< N, we have
|IV| ≤Cpr0
H1,rΩp
r0−1
H1,rΩ
vH1,NΩ. 4.15
Case 2rN.
|IV| ≤Cp−p0LqΩ∇p
r0−1
LrΩ∇vLqr/qr−r0q−rΩ r < q <∞. 4.16
Sinceqr/qr−r0 q−r< qr/q−r, we have
|IV| ≤Cpr0
H1,rΩp
r0−1
H1,rΩ
Case 3r > N.
|IV| ≤C∇pr0−1
LrΩ∇vLrΩ. 4.18
These estimates lead to
|I||II||III||IV| ≤FpH1,rΩ
vH1,σ∗Ω 4.19
for some polynomialF.
Proof ofTheorem 4.2. Using Theorem 3.2, let z ∈ H01,σΩ θ0,then for 3.2 there exists a
unique solutionpzsatisfying
pz
H1,rΩ≤C. 4.20
Moreover, if limm→ ∞zmza.e. inΩ, then
lim
m→ ∞pzm pz strongly inH
1,rΩ.
4.21
Next, usingLemma 4.3, we can define a linear functionalFz∈H1,σ ∗
Ω∗determined by
Fz, v
Ωkθ
∇pzr−2∇
pz· ∇p0v dx
−
Ωkθ
∇pzr−2∇
pz
pz−p0
· ∇v dx
−
Ωβ
∇pzr0−2∇pz· ∇
pz−p0
v dx
−
Ωβ
∇pzr0−2∇pz
pz−p0
· ∇v dx,
4.22
for allv ∈H1,σ∗Ω.By virtue of4.6,F
zis well defined, and there exists a constantC >0
independent ofzsuch that
|Fz, v| ≤CvH1,σ∗Ω. 4.23
We notice that 4.2 is the same as 11, equation 1.6 . Therefore, arguments after 11, equation3.19 can be used to complete the proof ofTheorem 4.2.
Acknowledgments
supported in part by NSF Grants OISE-0438765 and DMS-0920850. The project is also partially supported by a grant at Fudan University.
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