Nonlinear Analysis 102 (2014) 168–185

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Non-isothermal Navier–Stokes system with mixed boundaryconditions and friction law: Uniqueness andregularity propertiesMahdi Boukrouche ∗, Imane Boussetouan, Laetitia PaoliLyon University, UJM F-42023 Saint-Etienne, I.C.J CNRS UMR 5208, 23 rue Paul Michelon 42023 Saint-Etienne Cedex 2, France

a r t i c l e i n f o

Article history:Received 17 September 2013Accepted 14 February 2014Communicated by S. Carl

MSC:76D0576D0374M1035R3535A3535B45

Keywords:Unsteady Navier–Stokes systemTresca’s friction lawTemperature-dependent viscosityPenalty methodRegularity propertiesUniqueness

a b s t r a c t

Weconsider an unsteady non-isothermal fluid flow subjected to non-homogeneousDirich-let conditions on a part of the boundary and Tresca’s friction law on the other part. Forthis problem an existence result has been proved recently in Boukrouche et al. (2014) butuniqueness has been left as an open question. Starting from the approximation of the prob-lem based on a regularization of the free boundary condition due to friction combinedwitha special penalty method, we establish some sharp a priori estimates leading to better reg-ularity properties for the velocity field and to the uniqueness of the solutions. Finally westudy the regularity of the pressure field and of the stress tensor.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Motivated by lubrication and injection/extrusion phenomena where friction and heat transfer play an important role[1–4], we consider in this paper an unsteady non-isothermal incompressible fluid flow with non standard boundary condi-tions. The problem is described by the Navier–Stokes system with a temperature-dependent viscosity for the velocity andpressure fields and by the heat transfer equation for the temperature. If the thermal capacity and conductivity of the fluidare independent of the velocity and the pressure, which is usually true for this kind of applications, we may consider thetemperature as a data and we get the following system [5,6]:

∂v

∂t+ (v.∇)v − div(σ ) = f in Ω × (0, τ ), with σ = −pI + 2µ(T )D(v), (1.1)

div(v) = 0 in Ω × (0, τ ), (1.2)

∗ Corresponding author. Tel.: +33 477481535.E-mail addresses:Mahdi.Boukrouche@univ-st-etienne.fr (M. Boukrouche), Imane.Boussetouan@univ-st-etienne.fr (I. Boussetouan),

Laetitia.Paoli@univ-st-etienne.fr (L. Paoli).

http://dx.doi.org/10.1016/j.na.2014.02.0090362-546X/© 2014 Elsevier Ltd. All rights reserved.

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 169

with the initial condition

v(0, ·) = v0. (1.3)

Here T , v and p denote respectively the temperature, the velocity and the pressure andD(v) is the strain rate tensor given by

D(v) =dij(v)

1≤i,j≤d , dij(v) =

12

∂vi

∂xj+

∂vj

∂xi

1 ≤ i, j ≤ d.

The fluid domain Ω ⊂ Rd, d = 2, 3, is given by

Ω =(x′, xd) ∈ Rd

: x′= (x1, . . . , xd−1) ∈ ω, 0 < xd < h(x′)

,

where ω is a non empty open bounded subset of Rd−1 and h is a function of class C1 on Rd−1 which is bounded from aboveand from below by two positive real numbers. Let us emphasize that we do not introduce any restrictive assumption on thethickness of the domain so that we may consider either thin films (lubrication problems) or more general 3D geometries(injection/extrusion problems). The boundary ofΩ is decomposed as ∂Ω = Γ0∪Γ1∪ΓL, withΓ0 = (x′, xd) ∈ Ω : xd = 0,Γ1 = (x′, xd) ∈ Ω : xd = h(x′) andΓL the lateral boundary. The fluid velocity satisfies non-homogeneous Dirichlet bound-ary conditions on Γ1 ∪ ΓL and Tresca’s friction law on Γ0. More precisely let u · w be the Euclidean inner product of twovectors u and w and |u| be the Euclidean norm of u. We define the normal and tangential velocities on Γ0 by

vn = v · n = vini, vT =vT i1≤i≤d with vT i = vi − vnni 1 ≤ i ≤ d

and the normal and tangential components of the stress tensor on Γ0 by

σn = (σ · n) · n = σijnjni, σT =σT i1≤i≤d with σT i = σijnj − σnni 1 ≤ i ≤ d

where n = (n1, . . . , nd) is the unit outward normal vector to ∂Ω . Note that we will use Einstein’s summation conventionthroughout this paper. We assume that the upper part of the boundary is fixed. Moreover we denote by s : Γ0 → Rd−1 theshear velocity of the lower part of the boundary at t = 0 and by sζ (t), with ζ (0) = 1, its velocity at any instant t ∈ [0, τ ].We introduce a function g : ∂Ω → Rd such that

ΓL

g · n dγ = 0, g = 0 on Γ1, gn = g · n = 0 and gτ = g − gnn = s on Γ0.

Then non-homogeneous boundary conditions are prescribed on Γ1 ∪ ΓL with

v = 0 on Γ1 × (0, τ ), v = gζ on ΓL × (0, τ ), (1.4)

and Tresca’s friction law is given on Γ0 [7] i.e.

vn = v · n = 0 on Γ0 × (0, τ ), (1.5)

and

|σT | < ℓ ⇒ vT = sζ|σT | = ℓ ⇒ ∃λ ≥ 0 such that vT = sζ − λσT

(1.6)

where ℓ is the upper limit for the shear stress (i.e. ℓ is Tresca’s friction threshold).For this problem an existence result has been established in [8]: a sequence of approximate solutions (vδ

εm,

pδεm)ε>0,δ>0,m≥1 is constructed by using a regularization of Tresca’s friction law combined with a penalization of the di-

vergence free condition and Galerkin’s method, then a priori estimates and compactness arguments allow us to pass to thelimit and to get a solution. As usual with a convergence procedure, uniqueness cannot be derived as a by-product of theexistence proof. Our purpose is thus to complete the previous study and to prove uniqueness. Of course, due to Sobolev’sinjections, the space dimensionwill play an important role (see [9,6,10–12] for instance) andwewill obtain different resultsin the 2D and 3D cases.

More precisely, as for the classical problem of Navier–Stokes system with homogeneous Dirichlet boundary conditionsand constant viscosity, we will prove uniqueness in the 2D case, while wewill obtain uniqueness only in a restricted class of(more regular) solutions in the 3D case. This will lead us to go further in the study of the approximate solutions introducedin [8] and to establish additional a priori estimates yielding to better regularity properties for the limit, provided that sometechnical assumptions on the data are satisfied (roughly speaking the data should be small enough and/or the viscosityshould remain large enough).

The paper is organized as follows. In Section 2we introduce themathematical formulation of the problem as a variationalinequality. Then in Section 3, we derive a uniqueness result in the 2D case and we establish uniqueness for more regularsolutions in the 3D case. In Section 4 we recall the construction of the approximate solutions (vδ

εm, pδεm)ε>0,δ>0,m≥1 and we

prove new a priori estimates leading to better regularity properties for the limit velocity asm tends to+∞ and δ and ε tendto zero. Finally, in Section 5 we study the regularity of the pressure field and of the stress tensor.

170 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

2. Variational formulation of the problem

Let

H1(Ω) =H1(Ω)

d, L2(Ω) = (L2(Ω))d, L4(Ω) = (L4(Ω))d, H2(Ω) =

H2(Ω)

dand

L20(Ω) =

q ∈ L2(Ω) :

Ω

q dx = 0

.

As in [8] we define the functional spaces V0 and V0div by

V0 =ϕ ∈ H1(Ω) : ϕ = 0 on Γ1 ∪ ΓL, ϕn = 0 on Γ0

endowed with the norm of H1(Ω) and

V0div =ϕ ∈ V0 : div(ϕ) = 0 in Ω

.

Let µ ∈ C1(R, R) and assume that there exist three real numbers µ∗, µ∗ and µ′∗such that

0 < µ∗≤ 2µ(X) ≤ µ∗,

µ′(X) ≤ µ′

∗∀X ∈ R. (2.1)

For a given temperature field T ∈ L20, τ ; L2(Ω)

, with τ > 0, we define

a(T ; u, v) =

Ω

2µ(T )dij(u)dij(v) dx ∀(u, v) ∈ H1(Ω) × H1(Ω), ∀a.e. t ∈ (0, τ )

and we denote by b the trilinear form given by

b(u, v, w) =

Ω

ui∂vj

∂xiwj dx ∀(u, v, w) ∈ H1(Ω) × H1(Ω) × H1(Ω).

By definition of V0 we have the identity

b(u, v, w) = −b(u, w, v) −

Ω

div(u)v · w dx ∀(u, v, w) ∈ V0 × V0 × V0. (2.2)

Moreover, using Korn’s inequality [13], there exists α > 0 such that, for almost every t ∈ (0, τ ), we have

α∥u∥2H1(Ω)

≤ a(T ; u, u) ≤ µ∗∥u∥2H1(Ω)

∀u ∈ V0. (2.3)

We assume that

f ∈ L20, τ ; L2(Ω)

, ℓ ∈ L2

0, τ ; L2

+(Γ0)

∩ L∞

0, τ ; L∞

+(Γ0)

, (2.4)

where L2+(Γ0) = u ∈ L2(Γ0) : u ≥ 0 and L∞

+(Γ0) = u ∈ L∞(Γ0) : u ≥ 0,

ζ ∈ C∞[0, τ ]

with ζ (0) = 1. (2.5)

We introduce Tresca’s functional Ψ i.e.

Ψ : L20, τ ; L2(Γ0)

→ R

u → Ψ (u) =

τ

0

Γ0

ℓ|u| dx′dt.

Wemay observe that Ψ is convex continuous but not differentiable.In order to deal with homogeneous boundary conditions on Γ1 ∪ ΓL, we assume also that there exists an extension of g

to Ω , denoted by G0, such thatG0 ∈ H2(Ω), div(G0) = 0 in Ω,G0 = 0 on Γ1, G0 = g on ΓL, G0n = 0 and G0τ = s on Γ0

(2.6)

and we letv = v − G0ζ . The variational formulation of the problem (1.1)–(1.6) is given by (see for instance [7,14,15])

Problem (P). Findv ∈ L20, τ ; V0div

∩ L∞

0, τ ; L2(Ω)

, p ∈ H−10, τ ; L20(Ω)

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 171

such that, for all ϕ ∈ V0 and for all χ ∈ D(0, τ ), we have∂

∂t(v, ϕ) , χ

+b(v,v, ϕ), χ

−p, div(ϕ)

, χ+a(T ;v, ϕ), χ

+ Ψ (v + ϕχ) − Ψ (v)

≥(f , ϕ), χ

−ζa(T ;G0, ϕ), χ

−

∂ζ

∂t(G0, ϕ) , χ

−ζb(G0,v + G0ζ , ϕ), χ

−ζb(v,G0, ϕ), χ

(2.7)

with the initial conditionv(0, ·) = v0 − G0 =v0 ∈ L2(Ω) (2.8)

where (·, ·) denotes the inner product in L2(Ω) and ⟨·, ·⟩ = ⟨·, ·⟩D ′(0,τ ),D(0,τ ) the duality product between D ′(0, τ ) and thespace D(0, τ ) of functions of class C∞ with compact supports in (0, τ ). Let us emphasize that we identifyv + ϕχ andvwith their trace on Γ0 in the definition of Ψ (v + ϕχ) and Ψ (v).

For this problem an existence result has been established in [8]:

Theorem 2.1 (Theorem 6.3 in [8]). Assume that (2.1) and (2.4)–(2.6) hold and T ∈ L20, τ ; L2(Ω)

. Then Problem(P) admits

(at least) a solution (v, p). Furthermore ∂v∂t belongs to L

430, τ ; (V0div)

′.

The proof relies on the study of a sequence of approximate solutions whose convergence is obtained by means of com-pactness arguments. Of course with this kind of technique, uniqueness cannot be derived directly and was left as an openquestion.

3. Uniqueness properties

In the 2D case, the classical Sobolev’s inequality (see for instance [16])

∥u∥L4(Ω) ≤ C(Ω)∥u∥12L2(Ω)

∥u∥12H1(Ω)

∀u ∈ H1(Ω) (3.1)

allows us to improve the regularity of the solutions of Problem (P) and to prove uniqueness. Indeed

Lemma 3.1. Let us assume that (2.1) and (2.4)–(2.6) hold and T ∈ L20, τ ; L2(Ω)

. Let us assume moreover that d = 2. Then

for any solution (v, p) of Problem(P) we have ∂v∂t ∈ L2

0, τ ; (V0div)

′.

Proof. Starting from (2.2) we haveb(u, u, w) =

b(u, w, u) ≤ ∥u∥2

L4(Ω)∥w∥H1(Ω) ∀(u, w) ∈ V0div × V0div

and with (3.1)b(u, u, w) ≤ C2(Ω)∥u∥L2(Ω)∥u∥H1(Ω)∥w∥H1(Ω) ∀(u, w) ∈ V0div × V0div.

Let (v, p) be a solution of Problem (P). For all ϕ ∈ V0div and for all χ ∈ D(0, τ ), we have∂

∂t(v, ϕ) , χ

≥ −

b(v + G0ζ ,v + G0ζ , ϕ) − a(T ;v + G0ζ , ϕ), χ

+

(f , ϕ) −

∂ζ

∂t(G0, ϕ) , χ

+ Ψ (v) − Ψ (v + ϕχ)

and ∂

∂t(v, ϕ) , −χ

≥ −

b(v + G0ζ ,v + G0ζ , ϕ) − a(T ;v + G0ζ , ϕ),−χ

+

(f , ϕ) −

∂ζ

∂t(G0, ϕ) , −χ

+ Ψ (v) − Ψ (v − ϕχ).

But Ψ (v) − Ψ (v + ϕχ) ≤

τ

0

Γ0

ℓ |v| − |v + ϕχ |

dx′dt ≤

τ

0

Γ0

ℓ|ϕχ | dx′dt

and Ψ (v − ϕχ) − Ψ (v) ≤

τ

0

Γ0

ℓ |v − ϕχ | − |v|

dx′dt ≤

τ

0

Γ0

ℓ|ϕχ | dx′dt,

172 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

it follows that ∂

∂t(v, ϕ) , χ

≤

C2(Ω)∥v∥L∞(0,τ ;L2(Ω))∥v∥L2(0,τ ;H1(Ω)) + 2K 2

∥v∥L2(0,τ ;H1(Ω))∥G0ζ∥L∞(0,τ ;H1(Ω))

+ K 2∥G0ζ∥L2(0,τ ;H1(Ω))∥G0ζ∥L∞(0,τ ;H1(Ω)) + µ∗∥v + G0ζ∥L2(0,τ ;H1(Ω)) + ∥f ∥L2(0,τ ;L2(Ω))

+

∂ζ

∂t

L2(0,τ )

∥G0∥L2(Ω) + γ (Ω)∥ℓ∥L2(0,τ ;L2(Γ0))

∥ϕχ∥L2(0,τ ;H1(Ω)),

where γ (Ω) is the norm of the trace operator from H1(Ω) into L2(∂Ω) and K is the norm of the identity mapping fromH1(Ω) into L4(Ω). By density of D(0, τ ) ⊗ V0div into L2

0, τ ; V0div

, we may conclude that ∂v

∂t ∈ L20, τ ; (V0div)

′.

Proposition 3.1 (Uniqueness in the 2D Case). Let us assume that (2.1) and (2.4)–(2.6) hold and T ∈ L20, τ ; L2(Ω)

. Let us

assume moreover that d = 2. Then Problem(P) admits a unique solution.

Proof. Let (v1, p1) and (v2, p2) be two solutions of Problem (P). With the previous regularity property we may rewrite(2.7) as τ

0

∂v1

∂t, ϕχ

V′0div,V0div

dt +

τ

0b(v1 + G0ζ ,v1 + G0ζ , ϕχ) dt +

τ

0a(T ;v1 + G0ζ , ϕχ) dt

+

τ

0

Γ0

ℓ (|v1 + ϕχ | − |v1|) dx′dt ≥

τ

0(f , ϕχ) dt −

τ

0

∂ζ

∂t(G0, ϕχ) dt

for any ϕ ∈ V0div and χ ∈ D(0, τ ). By density of D(0, τ ) ⊗ V0div in L2(0, τ ; V0div) wemay replace ϕχ by (v2 −v1)1[0,s] forany s ∈ [0, τ ] and we get s

0

∂v1

∂t, (v2 −v1)

V′0div,V0div

dt +

s

0b(v1 + G0ζ ,v1 + G0ζ ,v2 −v1) dt

+

s

0a(T ;v1 + G0ζ ,v2 −v1) dt +

s

0

Γ0

ℓ (|v2| − |v1|) dx′dt

≥

s

0(f ,v2 −v1) dt −

s

0

∂ζ

∂t(G0,v2 −v1) dt. (3.2)

Similarly we have s

0

∂v2

∂t, (v2 −v1)

V′0div,V0div

dt +

s

0b(v2 + G0ζ ,v2 + G0ζ ,v1 −v2) dt

+

s

0a(T ;v2 + G0ζ ,v1 −v2) dt +

s

0

Γ0

ℓ (|v1| − |v2|) dx′dt

≥

s

0(f ,v1 −v2) dt −

s

0

∂ζ

∂t(G0,v1 −v2) dt. (3.3)

Adding (3.2) and (3.3) we obtain

12

s

0

∂

∂t∥v1 −v2∥

2L2(Ω)

dt + α

s

0∥v1 −v2∥

2H1(Ω)

dt

≤

s

0

b(v2 + G0ζ ,v2 + G0ζ ,v1 −v2) − b(v1 + G0ζ ,v1 + G0ζ ,v1 −v2)

dt,

and using

b(v1 + G0ζ ,v1 + G0ζ ,v1 −v2) = b(v1 −v2,v1 + G0ζ ,v1 −v2) + b(v2 + G0ζ ,v1 + G0ζ ,v1 −v2)

= b(v1 −v2,v1 + G0ζ ,v1 −v2) + b(v2 + G0ζ ,v1 −v2,v1 −v2)

+ b(v2 + G0ζ ,v2 + G0ζ ,v1 −v2)

= b(v1 −v2,v1 + G0ζ ,v1 −v2) + b(v2 + G0ζ ,v2 + G0ζ ,v1 −v2),

we get

12

s

0

∂

∂t∥v1 −v2∥

2L2(Ω)

dt + α

s

0∥v1 −v2∥

2H1(Ω)

dt ≤

s

0

−b(v1 −v2,v1 + G0ζ ,v1 −v2)

dt.

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 173

Asv1(0) =v2(0) =v0, we obtain

12∥v1(s) −v2(s)∥2

L2(Ω)+ α

s

0∥v1 −v2∥

2H1(Ω) dt

≤ −

s

0b(v1 −v2,v1 + G0ζ ,v1 −v2) dt

≤

s

0C2(Ω)∥v1 −v2∥L2(Ω)∥v1 −v2∥H1(Ω)∥v1 + G0ζ∥H1(Ω) dt

≤ α

s

0∥v1 −v2∥

2H1(Ω)

dt +C4(Ω)

4α

s

0∥v1 −v2∥

2L2(Ω)

∥v1 + G0ζ∥2H1(Ω)

dt. (3.4)

Since ∥v1 + G0ζ∥2H1(Ω)

belongs to L1(0, τ ; R+), we infer with Gronwall’s lemma that

∥v1(s) −v2(s)∥2L2(Ω)

≤ 0 ∀s ∈ [0, τ ].

Then, with (2.7) we havep1 − p2, div(ϕ)

, χ= 0 ∀ϕ ∈ H1

0(Ω), ∀χ ∈ D(0, τ ).

For any w ∈ L20(Ω) there exists ϕ ∈ H10(Ω) such that div(ϕ) = w (see [17] or [18]). Hence

(p1 − p2, w), χ= 0 ∀w ∈ L20(Ω), ∀χ ∈ D(0, τ ).

Now let w ∈ L2(Ω) and

w = w −1

|Ω|

Ω

w dx ∈ L20(Ω).

Then (p1 − p2, w), χ

=(p1 − p2, w), χ

+

1|Ω|

Ω

w dx

Ω

(p1 − p2) dx, χ

=(p1 − p2, w), χ

= 0.

By density of D(0, τ ) ⊗ L2(Ω) into H10

0, τ ; L2(Ω)

we get

p1 − p2, ηH−1(0,τ ;L2(Ω)),H1

0 (0,τ ;L2(Ω))= 0 ∀η ∈ H1

0

0, τ ; L2(Ω)

and thus p1 = p2.

Let us observe that these last two results rely on the estimate (3.1) which implies that for any solution of Problem (P),vbelongs to L4

0, τ ; L4(Ω)

. In the 3D case this argument is not anymore valid and it iswell known that the uniqueness for 3D

Navier–Stokes problems remains an openquestion. Partial results of uniqueness locally in time [10] or for special geometries,like for instance thin domains [19,14,15], are available but for general 3D geometries uniqueness may be expected only fora restricted class of more regular solutions [6,20]. More precisely, within the class of solutions (v, p) of Problem (P) suchthatv ∈ Lq

0, τ ; L4(Ω)

for some value of q ≥ 8, the proofs of Lemma 3.1 and Proposition 3.1 may be reproduced, leading

to a uniqueness property. Indeed, we may use the following Gagliardo–Nirenberg’s inequality [21]

∥u∥L4(Ω) ≤ C(Ω)∥u∥14L2(Ω)

∥u∥34H1(Ω)

∀u ∈ H1(Ω) (3.5)

instead of (3.1) and we may replace (3.4) by

12∥v1(s) −v2(s)∥2

L2(Ω)+ α

s

0∥v1 −v2∥

2H1(Ω) dt

≤ −

s

0b(v1 −v2,v1 + G0ζ ,v1 −v2) dt

=

s

0b(v1 −v2,v1 −v2,v1 + G0ζ ) dt

≤

s

0C(Ω)∥v1 −v2∥

14L2(Ω)

∥v1 −v2∥74H1(Ω)

∥v1 + G0ζ∥L4(Ω) dt

≤ α

s

0∥v1 −v2∥

2H1(Ω)

dt +18

78α

7

C8(Ω)

s

0∥v1 −v2∥

2L2(Ω)

∥v1 + G0ζ∥8L4(Ω)

dt

which allows us to conclude again. Hence we have

174 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

Proposition 3.2 (Uniqueness in the 3D Case). Let us assume that (2.1) and (2.4)–(2.6) hold and T ∈ L20, τ ; L2(Ω)

. Let us

assume moreover that d = 3. There exists at most one solution (v, p) of Problem(P) such thatv ∈ L80, τ ; L4(Ω)

.

4. Regularity properties for the velocity

As we have obtained a uniqueness result for a restricted class of more regular solutions in the 3D case, the next stepin our study consists in proving the existence of such more regular solutions to Problem (P). So we have to go back to theexistence proof proposed in [8] and get better insights into the properties of the solutions.

Let us recall briefly the strategy of the existence proof. As usual for such problems we have first introduced a regulariza-tion of Tresca’s functional Ψ . Namely, for any ε > 0, we consider the approximate Problem (Pε)

Problem (Pε). Findvε ∈ L20, τ ; V0div

∩ L∞

0, τ ; L2(Ω)

, pε ∈ H−10, τ ; L20(Ω)

such that, for all ϕ ∈ V0 and for all χ ∈ D(0, τ ), we have

∂

∂t(vε, ϕ) , χ

+b(vε,vε, ϕ), χ

−pε, div(ϕ)

, χ+a(T ;vε, ϕ), χ

+

τ

0

Γ0

ℓvε · ϕ

ε2 + |vε|2χ dx′dt =

(f , ϕ), χ

−ζa(T ;G0, ϕ), χ

−

∂ζ

∂t(G0, ϕ) , χ

−ζb(G0,vε + G0ζ , ϕ), χ

−ζb(vε,G0, ϕ), χ

(4.1)

with vε(0) =vε0 ∈ L2(Ω). (4.2)

In order to solve (Pε) wemay choose divergence free test-functions ϕ ∈ V0div: we get a variational problem for the fluidvelocityvε andwemay derive the pressure pε via abstract results of functional analysis (see [6,22,12] for instance).With thistechnique the pressure appears only as a by-product of the study. So we have chosen in [8] to follow an idea proposed byJ.L. Lions in [18] (see also [23]) and to build an approximate pressure by using a penalization of the divergence free conditioncombined with Galerkin’s method.

More precisely, since V0 is a closed subspace of H1(Ω), it admits an Hilbertian basis (wi)i≥1, which is orthogonal for theinner product of H1(Ω) and orthonormal for the inner product of L2(Ω). Then, for all ε > 0, δ > 0 and m ≥ 1, we look fora functionvδ

εm : [0, τ ] → Span(w1, . . . , wm) such that, for all k ∈ 1, . . . ,m, we have∂vδ

εm

∂t, wk

+ b(vδ

εm,vδεm, wk) +

12

vδεmdiv(vδ

εm), wk+

1δ

div(vδ

εm), div(wk)

+ a(T ;vδεm, wk) +

Γ0

ℓvδ

εm · wkε2 + |vδ

εm|2dx′

= (f , wk) − ζa(T ;G0, wk)

−∂ζ

∂t(G0, wk) − ζb(G0,vδ

εm + G0ζ , wk) − ζb(vδεm,G0, wk) in D ′(0, τ ) (4.3)

with vδεm(0, ·) =vδ

εm0 (4.4)

wherevδεm0 is defined as the orthogonal projection ofvδ

ε0 in L2(Ω) on Spanw1, . . . , wm

and we assume that the sequence

of initial data (vδε0)δ>0 satisfies

vδε0 −→δ→0vε0 −→ε→0v0 strongly in L2(Ω). (4.5)

The last term of the first line of (4.3) is the penalty term: − 1δdiv(vδ

εm) plays the role of an approximate pressure. Further-more the third term of the first line of (4.3) is introduced for technical reasons: indeed, with (2.2) we will have

b(vδεm,vδ

εm,vδεm) +

12

vδεmdiv(vδ

εm),vδεm

= 0.

We may rewrite (4.3)–(4.4) as a differential system and by using Caratheodory’s theorem [24] we prove the existence of aunique solution. Moreover

Lemma 4.1 (Lemma 4.1, [8]). Assume that (2.1), (2.4)–(2.6) hold and T ∈ L20, τ ; L2(Ω)

. Assume also that (vδ

ε0)ε>0,δ>0 is abounded sequence of L2(Ω). Then problem (4.3)–(4.4) admits a unique solutionvδ

εm ∈ W 1,20, τ ; Span(w1, . . . , wm)

which

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 175

satisfies the following estimates

∥vδεm∥L∞(0,τ ;L2(Ω)) ≤ C1 (4.6)

∥vδεm∥L2(0,τ ;H1(Ω)) ≤ C1 (4.7)div(vδ

εm)L2(0,τ ;L2(Ω))

≤ C1√

δ (4.8)

where C1 denotes a constant, independent of m, δ and ε.

Then the existence of a solution to Problem (P) is established via the convergence of the approximate velocities andpressures (vδ

εm, pδεm), where pδ

εm = −1δdiv(vδ

εm) (see [8]). In order to get better regularity properties for the limit velocitywe need additional a priori estimates for the sequence (vδ

εm)ε>0,δ>0,m≥1. More precisely, following the same ideas as in [6],

we will prove in Proposition 4.1 and Proposition 4.2 that

∂vδεm

∂t

ε>0,δ>0,m≥1

remains bounded in L20, τ ;H1(Ω)

and in

L∞0, τ ; L2(Ω)

provided that the data are more regular and also that they are ‘‘small’’ enough and/or that the viscosity is

large enough in the 3D case.So let us assume from now on that

f ∈ W 1,20, τ ; L2(Ω), ℓ ∈ W 1,20, τ ; L2

+(Γ0)

(4.9)

and

T ∈ W 1,∞0, τ ; L∞(Ω). (4.10)

Then, for all ε > 0, δ > 0 andm ≥ 1 we havevδεm ∈ W 2,2

0, τ ; Span(w1, . . . , wm)

and (4.3) is satisfied for all t ∈ [0, τ ].

Let us assume also the following compatibility conditions on the initial data

v0 ∈ H2(Ω), div(v0) = 0 in Ω, v0 = g on ∂Ω (4.11)

and

∂v0

∂xd= 0 on Γ0. (4.12)

Then we may choose G0 = v0: the initial condition (2.8) becomesv(0, ·) = v0 − G0 =v0 = 0 ∈ L2(Ω)

and we letvδε0 =vε0 =v0 for all ε > 0 and δ > 0.

It follows that

Lemma 4.2. Let us assume that (2.1), (2.4)–(2.6) and (4.9)–(4.12) hold. Assume also

T (0) ∈ W 1,4(Ω). (4.13)

Then, for all ε > 0, δ > 0 and m ≥ 1, we have∂vδεm

∂t(0)L2(Ω)

≤ A0,

where

A0 = K 2∥v0∥H1(Ω)∥v0∥H2(Ω) + 2K∥µ′

∥L∞(R)

∇T (0)

L4(Ω)∥v0∥H1(Ω)

+ µ∗∥v0∥H2(Ω) +f (0)L2(Ω)

+

∂ζ

∂t(0) ∥v0∥L2(Ω)

is independent of m, δ and ε.

Proof. Taking t = 0 in (4.3) we obtain∂vδ

εm

∂t(0), wk

= −b

vδεm(0) + G0ζ (0),vδ

εm(0) + G0ζ (0), wk−

12

vδεm(0)div

vδεm(0)

, wk

−

1δ

divvδ

εm(0), div(wk)

− a

T (0);vδ

εm(0) + G0ζ (0), wk

−

Γ0

ℓ(0)vδ

εm(0) · wkε2 + |vεm(0)|2

dx′+f (0), wk

−

∂ζ

∂t(0) (G0, wk) ∀1 ≤ k ≤ m.

176 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

Since ζ (0) = 1 andvδεm(0) =vδ

εm0 =v0 = 0 for allm ≥ 1, we get

12

vδεm(0)div

vδεm(0)

, wk

+

1δ

divvδ

εm(0), div(wk)

= 0 ∀1 ≤ k ≤ m

and Γ0

ℓ(0)vδ

εm(0) · wkε2 + |vεm(0)|2

dx′= 0 ∀1 ≤ k ≤ m.

Thus, reminding that G0 = v0,∂vδ

εm

∂t(0), w

= −b(v0, v0, w) − a

T (0); v0, w

+f (0), w

−

∂ζ

∂t(0) (v0, w) ∀w ∈ Span(w1, . . . , wm).

Now we choose w =∂vδ

εm∂t (0). By using Green’s formula we get∂vδ

εm

∂t(0)2L2(Ω)

= −b

v0, v0,∂vδ

εm

∂t(0)

+

Ω

2µ′T (0)

∂

∂xi

T (0)

dij(v0)

∂vjδεm

∂t(0) dx

+

Ω

2µ(T (0))

∂

∂xi

dij(v0)

∂vjδεm

∂t(0) dx +

f (0),

∂vδεm

∂t(0)

−∂ζ

∂t(0)

v0,

∂vδεm

∂t(0)

.

Let us estimate now the right-hand side of the previous equality. We havebv0, v0,∂vδ

εm

∂t(0) ≤ ∥v0∥L4(Ω)∥∇v0∥L4(Ω)

∂vδεm

∂t(0)L2(Ω)

≤ K 2∥v0∥H1(Ω)∥v0∥H2(Ω)

∂vδεm

∂t(0)L2(Ω)

,

where we recall that K denotes the norm of the identity mapping from H1(Ω) into L4(Ω). Using (2.1) we get2

Ω

µ′T (0)

∂

∂xi

T (0)

dij(v0)

∂vjδεm

∂t(0)dx

≤ 2∥µ′∥L∞(R)

∇T (0)L4(Ω)

∥∇v0∥L4(Ω)

∂vδεm

∂t(0)L2(Ω)

≤ 2K∥µ′∥L∞(R)

∇T (0)L4(Ω)

∥v0∥H1(Ω)

∂vδεm

∂t(0)L2(Ω)

and 2

Ω

µT (0)

∂

∂xi

dij(v0)

∂vjδεm

∂t(0)dx

≤ µ∗∥v0∥H2(Ω)

∂vδεm

∂t(0)L2(Ω)

,

it follows that∂vδεm

∂t(0)L2(Ω)

≤ A0 = K 2∥v0∥H1(Ω)∥v0∥H2(Ω) + 2K∥µ′

∥L∞(R)

∇T (0)

L4(Ω)∥v0∥H1(Ω)

+ µ∗∥v0∥H2(Ω) +f (0)L2(Ω)

+

∂ζ

∂t(0) ∥v0∥L2(Ω).

This ends the proof of this lemma.

We infer that

Proposition 4.1. Let us assume that (2.1), (2.4)–(2.6) and (4.9)–(4.13) hold. Let us assume also that d = 2.We have the followinga priori estimates∂vδ

εm

∂t

L∞(0,τ ;L2(Ω))

≤ C (4.14)∂vδεm

∂t

L2(0,τ ;H1(Ω))

≤ C (4.15)div∂vδεm

∂t

L2(0,τ ;L2(Ω))

≤ C√

δ (4.16)

where C is a constant independent of m, δ and ε.

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 177

Proof. Sincevδεm ∈ W 2,2

0, τ ; Span(w1, . . . , wm)

wemay differentiate all the terms of (4.3) with respect to the time vari-

able and we obtain∂2vδ

εm

∂t2, w

+ b

∂vδ

εm

∂t,vδ

εm, w

+ b

vδεm,

∂vδεm

∂t, w

+

12

∂vδ

εm

∂tdiv(vδ

εm), w

+

12

vδεmdiv

∂vδ

εm

∂t

, w

+

1δ

div

∂vδ

εm

∂t

, div(w)

+ a

T ;

∂vδεm

∂t, w

+

Ω

2µ′(T )∂T∂t

dij(vδεm)dij(w) dx +

∂

∂t

Γ0

ℓvδ

εm · wε2 + |vδ

εm|2dx′

=

∂ f∂t

, w

−

∂ζ

∂ta(T ;G0, w) − ζ

Ω

2µ′(T )∂T∂t

dij(G0)dij(w) dx −∂2ζ

∂t2(G0, w)

−∂ζ

∂tb(G0,vδ

εm + G0ζ , w) − ζbG0,

∂vδεm

∂t+ G0

∂ζ

∂t, w

−

∂ζ

∂tb(vδ

εm,G0, w) − ζb

∂vδεm

∂t,G0, w

a.e. in (0, τ )

for all w ∈ Span(w1, . . . , wm). Now we choose w =∂vδ

εm∂t . Observing that

∂

∂t

Γ0

ℓvδ

εm · wε2 + |vδ

εm|2dx′

=

Γ0

ℓ∂

∂t

vδεm

ε2 + |vδεm|2

· w dx′

+

Γ0

∂ℓ

∂tvδ

εm · wε2 + |vδ

εm|2dx′

and Γ0

ℓ∂

∂t

vδεm

ε2 + |vδεm|2

·∂vδ

εm

∂tdx′

=

Γ0

ℓ

∂vδ

εm∂t

2ε2 + |vδ

εm|2−

vδεm ·

∂vδεm

∂t

2ε2 + |vδ

εm|23/2

dx′

≥

Γ0

ℓε2 |∂vδ

εm∂t |

2ε2 + |vδ

εm|2 3

2dx′

≥ 0,

we get

12

∂

∂t

∂vδεm

∂t

2L2(Ω)

+ α

∂vδεm

∂t

2H1(Ω)

+1δ

div∂vδεm

∂t

2L2(Ω)

≤ −

Γ0

∂ℓ

∂tvδ

εm ·∂vδ

εm∂t

ε2 + |vδεm|2

dx′− b

∂vδ

εm

∂t,vδ

εm,∂vδ

εm

∂t

− b

vδεm,

∂vδεm

∂t,∂vδ

εm

∂t

−

12

∂vδ

εm

∂tdiv(vδ

εm),∂vδ

εm

∂t

−

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

−

Ω

2µ′(T )∂T∂t

dij(vδεm)dij

∂vδ

εm

∂t

dx

+

∂ f∂t

,∂vδ

εm

∂t

−

∂ζ

∂taT ;G0,

∂vδεm

∂t

− ζ

Ω

2µ′(T )∂T∂t

dij(G0)dij

∂vδ

εm

∂t

dx

−∂2ζ

∂t2

G0,

∂vδεm

∂t

−

∂ζ

∂tbG0,vδ

εm + G0ζ ,∂vδ

εm

∂t

− ζb

G0,

∂vδεm

∂t+ G0

∂ζ

∂t,∂vδ

εm

∂t

−

∂ζ

∂tbvδ

εm,G0,∂vδ

εm

∂t

− ζb

∂vδ

εm

∂t,G0,

∂vδεm

∂t

.

With relation (2.2) we have

bvδ

εm,∂vδ

εm

∂t,∂vδ

εm

∂t

+

12

∂vδ

εm

∂tdiv(vδ

εm),∂vδ

εm

∂t

= 0, and b

G0,

∂vδεm

∂t,∂vδ

εm

∂t

= 0.

Thus we get

12

∂

∂t

∂vδεm

∂t

2L2(Ω)

+ α

∂vδεm

∂t

2H1(Ω)

+1δ

div∂vδεm

∂t

2L2(Ω)

≤ −

Γ0

∂ℓ

∂tvδ

εm ·∂vδ

εm∂t

ε2 + |vδεm|2

dx′− b

∂vδ

εm

∂t,vδ

εm,∂vδ

εm

∂t

−

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

178 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

−

Ω

2µ′(T )∂T∂t

dij(vδεm)dij

∂vδ

εm

∂t

dx +

∂ f∂t

,∂vδ

εm

∂t

−

∂ζ

∂taT ;G0,

∂vδεm

∂t

− ζ

Ω

2µ′(T )∂T∂t

dij(G0)dij

∂vδ

εm

∂t

dx −

∂2ζ

∂t2

G0,

∂vδεm

∂t

−

∂ζ

∂tbG0,vδ

εm,∂vδ

εm

∂t

− 2ζ

∂ζ

∂tbG0,G0,

∂vδεm

∂t

−

∂ζ

∂tbvδ

εm,G0,∂vδ

εm

∂t

− ζb

∂vδ

εm

∂t,G0,

∂vδεm

∂t

. (4.17)

Let us estimate now the right side of inequality (4.17) by using Cauchy–Schwarz’s and Young’s inequalities. We obtain

Γ0

∂ℓ

∂tvδ

εm ·∂vδ

εm∂t

ε2 + |vδεm|2

dx′

≤ γ (Ω)

∂ℓ

∂t

L2(Γ0)

∂vδεm

∂t

H1(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7γ (Ω)2

2α

∂ℓ

∂t

2L2(Γ0)

,

where we recall that γ (Ω) is the norm of the trace operator from H1(Ω) to L2(Γ0),Ω

2µ′(T )∂T∂t

dij(vδεm)dij

∂vδ

εm

∂t

dx ≤ 2∥µ′

∥L∞(R)

∂T∂t

L∞(Ω)

∥∇vδεm∥L2(Ω)

∇ ∂vδεm

∂t

L2(Ω)

≤ c1∥vδεm∥H1(Ω)

∂vδεm

∂t

H1(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7c212α

∥vδεm∥

2H1(Ω)

with

c1 = 2∥µ′∥L∞(R)

∂T∂t

L∞(0,τ ;L∞(Ω))

, (4.18)

∂ f∂t

,∂vδ

εm

∂t

≤

∂ f∂t

L2(Ω)

∂vδεm

∂t

L2(Ω)

≤

∂ f∂t

2L2(Ω)

+14

∂vδεm

∂t

2L2(Ω)

,∂ζ

∂taT ;G0,

∂vδεm

∂t

≤ µ∗

∂ζ

∂t

∥G0∥H1(Ω)

∂vδεm

∂t

H1(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7µ2

∗

2α

∂ζ

∂t

2 ∥G0∥2H1(Ω)

,ζ Ω

2µ′(T )∂T∂t

dij(G0)dij

∂vδ

εm

∂t

dx ≤ 2∥µ′

∥L∞(R)|ζ |

∂T∂t

L∞(Ω)

∥G0∥H1(Ω)

∂vδεm

∂t

H1(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7c212α

|ζ |2∥G0∥

2H1(Ω)

,

∂2ζ

∂t2

G0,

∂vδεm

∂t

≤

∂2ζ

∂t2

∥G0∥L2(Ω)

∂vδεm

∂t

L2(Ω)

≤

∂2ζ

∂t2

2 ∥G0∥2L2(Ω)

+14

∂vδεm

∂t

2L2(Ω)

,∂ζ

∂tbG0,vδ

εm,∂vδ

εm

∂t

≤

∂ζ

∂t

∥G0∥L4(Ω)∥∇vδεm∥L2(Ω)

∂vδεm

∂t

L4(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7K 4

2α

∂ζ

∂t

2 ∥G0∥2H1(Ω)

∥vδεm∥

2H1(Ω)

,2ζ ∂ζ

∂tbG0,G0,

∂vδεm

∂t

≤ 2|ζ |

∂ζ

∂t

∥G0∥L4(Ω)∥∇G0∥L4(Ω)

∂vδεm

∂t

L2(Ω)

≤ 4K 4|ζ |

2∂ζ

∂t

2 ∥G0∥2H1(Ω)

∥G0∥2H2(Ω)

+14

∂vδεm

∂t

2L2(Ω)

,∂ζ

∂tbvδ

εm,G0,∂vδ

εm

∂t

≤

∂ζ

∂t

∥vδεm∥L4(Ω)∥∇G0∥L4(Ω)

∂vδεm

∂t

L2(Ω)

≤ K 4∂ζ

∂t

2 ∥G0∥2H2(Ω)

∥vδεm∥

2H1(Ω)

+14

∂vδεm

∂t

2L2(Ω)

,

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 179ζb∂vδεm

∂t,G0,

∂vδεm

∂t

≤ |ζ |

∂vδεm

∂t

L4(Ω)

∥∇G0∥L4(Ω)

∂vδεm

∂t

L2(Ω)

≤α

14

∂vδεm

∂t

2H1(Ω)

+7K 4

2α|ζ |

2∥G0∥

2H2(Ω)

∂vδεm

∂t

2L2(Ω)

,

where we recall that K is the norm of the identity mapping from H1(Ω) into L4(Ω).It remains to estimate the trilinear terms on the right-hand side of (4.17) i.e.

−b

∂vδεm

∂t,vδ

εm,∂vδ

εm

∂t

−

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

= b

∂vδ

εm

∂t,∂vδ

εm

∂t,vδ

εm

+

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

.

We will use here the space dimension. More precisely we use the inequality (3.1) i.e.

∥u∥L4(Ω) ≤ C(Ω)∥u∥12L2(Ω)

∥u∥12H1(Ω)

∀u ∈ H1(Ω)

which is valid in the 2D case and we getb∂vδεm

∂t,∂vδ

εm

∂t,vδ

εm

+

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

≤ c2

∂vδεm

∂t

12

L2(Ω)

∂vδεm

∂t

32

H1(Ω)

∥vδεm∥L4(Ω)

with c2 =3C(Ω)

2 . Then we apply Young’s inequality with (p, q) = 43 , 4

, we obtainb∂vδ

εm

∂t,vδ

εm,∂vδ

εm

∂t

+

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

≤

2α21

34∂vδ

εm

∂t

32

H1(Ω)

c2

212α

34

∥vδεm∥L4(Ω)

∂vδεm

∂t

12

L2(Ω)

≤

α

14

∂vδεm

∂t

2H1(Ω)

+14

212α

3

c42∥vδεm∥

4L4(Ω)

∂vδεm

∂t

2L2(Ω)

.

Gathering all the previous estimates, we infer

12

∂

∂t

∂vδεm

∂t

2L2(Ω)

+α

2

∂vδεm

∂t

2H1(Ω)

+1δ

div∂vδεm

∂t

2L2(Ω)

≤ A1 + A2∥vδεm∥

2H1(Ω)

+A3 + A4∥vδ

εm∥4L4(Ω)

∂vδεm

∂t

2L2(Ω)

a.e. in (0, τ ) (4.19)

with

A1 =7γ (Ω)2

2α

∂ℓ

∂t

2L2(Γ0)

+

∂ f∂t

2L2(Ω)

+ A′

1

A′

1 =7µ2

∗

2α

∂ζ

∂t

2C([0,τ ])

∥G0∥2H1(Ω)

+7c212α

∥ζ∥2C([0,τ ])∥G0∥

2H1(Ω)

+

∂2ζ

∂t2

2C([0,τ ])

∥G0∥2L2(Ω)

+ 4K 4∥ζ∥

2C([0,τ ])

∂ζ

∂t

2C([0,τ ])

∥G0∥2H1(Ω)

∥G0∥2H2(Ω)

,

A2 =7c212α

+7K 4

2α

∂ζ

∂t

2C([0,τ ])

∥G0∥2H1(Ω)

+ K 4∂ζ

∂t

2C([0,τ ])

∥G0∥2H2(Ω)

,

A3 = 1 +7K 4

2α∥ζ∥

2C([0,τ ])∥G0∥

2H2(Ω)

and

A4 =14

212α

3

c42 .

180 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

Let s ∈ [0, τ ]. By integration we have∂vδεm

∂t(s)2L2(Ω)

+ α

s

0

∂vδεm

∂t

2H1(Ω)

dt +2δ

s

0

div∂vδεm

∂t

2L2(Ω)

dt ≤

∂vδεm

∂t(0)2L2(Ω)

+ 2 s

0

A1 + A2∥vδ

εm∥2H1(Ω)

dt + 2

s

0

A3 + A4∥vδ

εm∥4L4(Ω)

∂vδεm

∂t

2L2(Ω)

dt

≤ A5 + 2 s

0

A3 + A4∥vδ

εm∥4L4(Ω)

∂vδεm

∂t

2L2(Ω)

dt (4.20)

with

A5 = A20 +

7γ (Ω)2

α

∂ℓ

∂t

2L2(0,τ ;L2(Γ0))

+ 2∂ f

∂t

2L2(0,τ ;L2(Ω))

+ 2A′

1τ + A2∥vδεm∥

2L2(0,τ ;H1(Ω))

, (4.21)

where A0 is the constant defined at Lemma 4.2. With Gronwall’s lemma we infer∂vδεm

∂t(s)2L2(Ω)

≤ A5 exp2 s

0

A3 + A4∥vδ

εm∥4L4(Ω)

dt

∀s ∈ [0, τ ].

By using again (3.1), we get∂vδεm

∂t(s)2L2(Ω)

≤ A5 exp2 s

0

A3 + A4C(Ω)4∥vδ

εm∥2L2(Ω)

∥vδεm∥

2H1(Ω)

dt

≤ A5 exp2A3τ + A4C(Ω)4∥vδ

εm∥2L∞(0,τ ;L2(Ω))

∥vδεm∥

2L2(0,τ ;H1(Ω))

∀s ∈ [0, τ ].

Then Lemma 4.1 implies that there exists a constant C , independent ofm, δ and ε such that∂vδεm

∂t(s)2L2(Ω)

≤ C ∀s ∈ [0, τ ]

and (4.14) follows immediately. Finally, by choosing s = τ in (4.20), we obtain (4.15) and (4.16).

By combining now Propositions 3.1 and 4.1 we obtain

Theorem 4.1 (Regularity and Uniqueness in the 2D Case). Let us assume that (2.1), (2.4)–(2.6) and (4.9)–(4.13) hold. Let usassume also that d = 2. Then Problem(P) admits a unique solution (v, p). Furthermore ∂v

∂t ∈ L20, τ ; V0div

∩ L∞

0, τ ; L2(Ω)

.

Let us consider now the 3D case. We may observe that in Proposition 4.1 all the computations remain valid except theestimate of the trilinear terms. Indeed, we still haveb∂vδ

εm

∂t,∂vδ

εm

∂t,vδ

εm

+

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

≤32

∂vδεm

∂t

L4(Ω)

∂vδεm

∂t

H1(Ω)

∥vδεm∥L4(Ω),

but we cannot use anymore the relation (3.1). Nevertheless, using the injection ofH1(Ω) into L4(Ω), which holds also whend = 3, we haveb∂vδ

εm

∂t,∂vδ

εm

∂t,vδ

εm

+

12

vδεmdiv

∂vδ

εm

∂t

,∂vδ

εm

∂t

≤32K 2∂vδ

εm

∂t

2H1(Ω)

∥vδεm∥H1(Ω),

where we recall once again that K is the norm of the identity mapping from H1(Ω) into L4(Ω). Then (4.19) becomes

12

∂

∂t

∂vδεm

∂t

2L2(Ω)

+12

α − 3K 2

∥vδεm∥H1(Ω)

∂vδεm

∂t

2H1(Ω)

+1δ

div∂vδεm

∂t

2L2(Ω)

≤ A1 + A2∥vδεm∥

2H1(Ω)

+ A3

∂vδεm

∂t

2L2(Ω)

a.e. in (0, τ ). (4.22)

Then we have to check the sign of α − 3K 2∥vδ

εm∥H1(Ω).

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 181

Proposition 4.2. Let us assume that (2.1), (2.4)–(2.6), (4.9)–(4.10) and the compatibility conditions on the initial data(4.11)–(4.13) hold. Let us assume also that d = 3. Let D be defined as

D = 2f (0)2L2(Ω)

+ 2τ∂ f

∂t

2L2(0,τ ;L2(Ω))

+2µ2

∗

α∥ζ∥

2C([0,τ ])∥G0∥

2H1(Ω)

+

∂ζ

∂t

2C([0,τ ])

∥G0∥2L2(Ω)

+ K 4∥ζ∥

4C([0,τ ])∥G0∥

2H1(Ω)

∥G0∥2H2(Ω)

+

4 +

2K 4

α∥ζ∥

2C([0,τ ])∥G0∥

2H2(Ω)

C21A5 exp(2A3τ), (4.23)

where C1 is the constant defined at Lemma 4.1 and assume that

D <α3

9K 4. (4.24)

Then

α − 3K 2vδ

εm(t)H1(Ω)

≥ α − 3K 2

Dα

> 0 ∀t ∈ [0, τ ].

Moreover we have the following a priori estimates∂vδεm

∂t

L∞(0,τ ;L2(Ω))

≤ C (4.25)∂vδεm

∂t

L2(0,τ ;H1(Ω))

≤ C (4.26)div∂vδεm

∂t

L2(0,τ ;L2(Ω))

≤ C√

δ (4.27)

where C is a constant independent of m, δ and ε.

Proof. With (4.3) we have∂vδ

εm

∂t(t), w

+ b

vδεm(t),vδ

εm(t), w+

12

vδεm(t)div(vδ

εm)(t), w+

1δ

div(vδ

εm)(t), div(w)

+ aT (t);vδ

εm(t), w+

Γ0

ℓvδ

εm(t) · wε2 + |vδ

εm(t)|2dx′

=f (t), w

− ζ (t)a

T (t);G0, w

−

∂ζ

∂t(t) (G0, w)

− ζ (t)bG0,vδ

εm(t) + G0ζ (t), w− ζ (t)b

vδεm(t),G0, w

∀t ∈ [0, τ ]

for all w ∈ Span(w1, . . . , wm). We choose w =vδεm(t). Recalling (2.2), we obtain

∂vδεm

∂t(t),vδ

εm(t)

+1δ

div(vδ

εm)(t), div(vδεm)(t)

+ a

T (t);vδ

εm(t),vδεm(t)

+

Γ0

ℓ

vδεm(t)

2ε2 + |vδ

εm(t)|2dx′

=f (t),vδ

εm(t)− ζ (t)a

T (t);G0,vδ

εm(t)

−∂ζ

∂t(t)G0,vδ

εm(t)−ζ (t)

2bG0,G0,vδ

εm(t)− ζ (t)b

vδεm(t),G0,vδ

εm(t)

∀t ∈ [0, τ ].

With (2.3) we infer

1δ

div(vδεm)(t)

2L2(Ω)

+ αvδ

εm(t)2H1(Ω)

≤f (t)L2(Ω)

vδεm(t)

L2(Ω)

+ µ∗

ζ (t)∥G0∥H1(Ω)

vδεm(t)

H1(Ω)

+

∂ζ

∂t(t) ∥G0∥L2(Ω)

vδεm(t)

L2(Ω)

+ζ (t)

2∥G0∥L4(Ω)∥∇G0∥L4(Ω)

vδεm(t)

L2(Ω)

+ζ (t)

vδεm(t)

L4(Ω)

∥∇G0∥L4(Ω)

vδεm(t)

L2(Ω)

+

∂vδεm

∂t(t)L2(Ω)

vδεm(t)

L2(Ω)

182 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

and with Young’s inequalities

1δ

div(vδεm)(t)

2L2(Ω)

+α

2

vδεm(t)

2H1(Ω)

≤12

f (t)2L2(Ω)+

µ2∗

α

ζ (t)2∥G0∥

2H1(Ω)

+12

∂ζ

∂t(t)2 ∥G0∥

2L2(Ω)

+K 4

2

ζ (t)4∥G0∥

2H1(Ω)

∥G0∥2H2(Ω)

+

2 +

K 4

α

ζ (t)2∥G0∥

2H2(Ω)

vδεm(t)

2L2(Ω)

+12

∂vδεm

∂t(t)2L2(Ω)

(4.28)

for all t ∈ [0, τ ]. For t = 0, using Lemmas 4.1 and 4.2, we get

αvδ

εm(0)2H1(Ω)

+2δ

div(vδεm)(0)

2L2(Ω)

≤f (0)2L2(Ω)

+2µ2

∗

α

ζ (0)2∥G0∥

2H1(Ω)

+

∂ζ

∂t(0)2 ∥G0∥

2L2(Ω)

+ K 4ζ (0)

4∥G0∥2H1(Ω)

∥G0∥2H2(Ω)

+

4 +

2K 4

α

ζ (0)2∥G0∥

2H2(Ω)

C21 + A2

0.

With the definition (4.23) of D and assumption (4.24), we obtain

α − 3K 2vδ

εm(0)H1(Ω)

> 0.

Sincevδεm ∈ W 1,2(0, τ ; V0) ⊂ C

[0, τ ];H1(Ω)

, we infer that α−3K 2

vδεm(t)

H1(Ω)

remains positive on a right neighbour-hood of 0. We define

τ δεm = sup

t ∈ [0, τ ]; α − 3K 2

vδεm(s)

H1(Ω)

> 0 ∀s ∈ [0, t]

> 0.

Then α − 3K 2∥vδ

εm(t)∥H1(Ω) ≥ 0 for all t ∈0, τ δ

εm

and

∂

∂t

∂vδεm

∂t

2L2(Ω)

≤ 2A1 + 2A2∥vδεm∥

2H1(Ω)

+ 2A3

∂vδεm

∂t

2L2(Ω)

a.e. in0, τ δ

εm

.

Similarly as in Proposition 4.1 we integrate and we use Gronwall’s lemma. We obtain∂vδεm

∂t(s)2L2(Ω)

≤ A5 exp(2A3s) ∀t ∈ [0, τ δεm].

Going back to (4.28) we infer that

2δ

div(vδεm)(t)

2L2(Ω)

+ αvδ

εm(t)2H1(Ω)

≤f (t)2L2(Ω)

+2µ2

∗

α

ζ (t)2∥G0∥

2H1(Ω)

+

∂ζ

∂t(t)2 ∥G0∥

2L2(Ω)

+ K 4ζ (t)

4∥G0∥2H1(Ω)

∥G0∥2H2(Ω)

+

4 +

2K 4

α

ζ (t)2∥G0∥

2H2(Ω)

C21

+ A5 exp(2A3t) ∀t ∈0, τ δ

εm

.

As f ∈ W 1,2(0, τ ; L2(Ω)), we have

f (t) = f (0) +

t

0

∂ f∂t

(s) ds ∀t ∈ [0, τ ]

thus f (t)2L2(Ω)≤ 2

f (0)2L2(Ω)+ 2τ

∂ f∂t

2L2(0,τ ;L2(Ω))

.

It follows that for all t ∈0, τ δ

εm

we have

αvδ

εm(t)2H1(Ω)

≤ D.

With (4.24) we get

α − 3K 2vδ

εm(t)H1(Ω)

≥ α − 3K 2

Dα

> 0 ∀t ∈0, τ δ

εm

and we may conclude that τ δ

εm = τ .

M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185 183

Now let s ∈ [0, τ ]. By an integration of (4.22) on [0, s] we get∂vδεm

∂t(s)2L2(Ω)

+

α − 3K 2

Dα

s

0

∂vδεm

∂t

2H1(Ω)

dt +2δ

s

0

div∂vδεm

∂t

2L2(Ω)

dt

≤ A5 + 2A3

s

0

∂vδεm

∂t

2L2(Ω)

dt

and it follows that∂vδεm

∂t(s)2L2(Ω)

≤ A5 exp(2A3s) ≤ A5 exp(2A3τ) ∀s ∈ [0, τ ].

With the definition of A5 (4.21) and Lemma 4.1 we obtain (4.25). Then, with s = τ , we get (4.26) and (4.27).

As a corollary we obtain

Theorem 4.2 (Regularity in the 3D Case). Let us assume that (2.1), (2.4)–(2.6), (4.9)–(4.10) and the compatibility conditions onthe initial data (4.11)–(4.13) hold. Let us assume also that d = 3 and

D <α3

9K 4

with D given by (4.23). Then Problem(P) admits (at least) a solution (v, p) such that ∂v∂t ∈ L2

0, τ ; V0div

∩ L∞

0, τ ; L2(Ω)

.

Remark 4.1. By the injection of W 1,2(0, τ ; V0div) into L∞0, τ ;H1(Ω)

⊂ L8

0, τ ; L4(Ω)

we may conclude with Proposi-

tion 3.2 such that a solution is unique.

Remark 4.2. Let us recall the expression of C1 (see Lemma 4.1 in [8]):

C1 =

max

2,

2α

C ′

1 exp(2C′′

1 τ)

12

with

C ′

1 =12∥f ∥2

L2(0,τ ;L2(Ω))+

µ2∗

α∥G0∥

2H1(Ω)

∥ζ∥2L2(0,τ )

+12∥G0∥

2L2(Ω)

∂ζ

∂t

2L2(0,τ )

+K 4

2∥G0∥

2H1(Ω)

∥∇G0∥2H1(Ω)

∥ζ∥4L4(0,τ )

and

C ′′

1 =32

+K 4

α∥∇G0∥

2H1(Ω)

∥ζ∥2L∞(0,τ ).

Since α is proportional to theminimum of the viscosity, the condition (4.24) means that the viscosity is large enough and/orthe data are ‘‘small’’ enough.

5. Regularity properties for the stress tensor

Let us consider in this section the solutions (v, p) of Problem (P) obtained as the limits of the sequence (vδεm,

pδεm)ε>0,δ>0,m≥1 (see [8]). Let us recall that the stress tensor σ is defined by

σ = (σij)1≤i,j≤d, σij = −pδij + 2µ(T )dij(v + G0ζ ) 1 ≤ i, j ≤ d.We obtain

Proposition 5.1. Let us assume that (2.1), (2.4)–(2.6), (4.9)–(4.10) and the compatibility conditions on the initial data(4.11)–(4.13) hold. Let us assume also that

D <α3

9K 4

with D given by (4.23) whenever d = 3. Then p ∈ L20, τ ; L20(Ω)

, σ ∈ L2

0, τ ;

L2(Ω)

d×dand div(σ ) ∈ L2

0, τ ; L

43 (Ω)

.

Proof. The first part of the result is an immediate consequence of Proposition 4.1 or Proposition 4.2. Indeed, the injectionofW 1,2

0, τ ; V0div

into L∞

0, τ ;H1(Ω)

implies that there exists a constant C independent ofm, δ and ε such that

∥vδεm∥L∞(0,τ ;H1(Ω)) ≤ C .

Then by using the same kind of computations as in Lemma 5.1 in [8] (with obvious changes due to the improved estimatesof the velocity) we may conclude that p ∈ L2

0, τ ; L20(Ω)

and σ ∈ L2

0, τ ;

L2(Ω)

d×d. Now let us choose ϕ ∈

D(Ω)

d

184 M. Boukrouche et al. / Nonlinear Analysis 102 (2014) 168–185

and χ ∈ D(0, τ ). With (2.7) we have∂

∂t(v, ϕ) , χ

+b(v + G0ζ ,v + G0ζ , ϕ), χ

−p, div(ϕ)

, χ

+a(T ;v + G0ζ , ϕ), χ

≥(f , ϕ), χ

−

∂ζ

∂t(G0, ϕ) , χ

and

∂

∂t(v, ϕ) , −χ

+b(v + G0ζ ,v + G0ζ , ϕ),−χ

−p, div(ϕ)

, −χ

+a(T ;v + G0ζ , ϕ),−χ

≥(f , ϕ),−χ

−

∂ζ

∂t(G0, ϕ) , −χ

.

It follows that∂

∂t(v, ϕ) , χ

+b(v + G0ζ ,v + G0ζ , ϕ), χ

+

τ

0

Ω

σij∂ϕi

∂xjχ dxdt

=(f , ϕ), χ

−

∂ζ

∂t(G0, ϕ) , χ

(5.1)

and thus τ

0

Ω

σij∂ϕi

∂xjχ dxdt

≤

∂v∂t

L2(0,τ ;L2(Ω))

+ ∥f ∥L2(0,τ ;L2(Ω)) +

∂ζ

∂t

L2(0,τ )

∥G0∥L2(Ω)

∥ϕχ∥L2(0,τ ;L2(Ω))

+√

τK∥v + G0ζ∥L∞(0,τ ;H1(Ω))∥ϕχ∥L2(0,τ ;L4(Ω))

≤

|Ω|

14

∂v∂t

L2(0,τ ;L2(Ω))

+ ∥f ∥L2(0,τ ;L2(Ω)) +

∂ζ

∂t

L2(0,τ )

∥G0∥L2(Ω)

+√

τK∥v + G0ζ∥L∞(0,τ ;H1(Ω))

∥ϕχ∥L2(0,τ ;L4(Ω)).

Hence div(σ ) ∈L20, τ ; L4(Ω)

′= L2

0, τ ; L

43 (Ω)

.

Conclusion

Since the work of the pioneering of J. Leray [5], the mathematical analysis of the Navier–Stokes system has madesignificant progress in many directions (see for instance [18,17,6,19,25,20,26–29,12,30–34]). In order to put this work inrelation to similar problems already discussed in the literature, let us point the relevant question of the behaviour for largetime in the 2D case [28,34]. For other boundary conditions, let us mention [28] also in the 2D case with mixed nonstandardboundary conditions and time dependent driving forces or [35] where very weak solutions of Navier–Stokes equations inexterior domains with nonhomogeneous data are studied.

Let us emphasize that uniqueness, for 3D Navier–Stokes problem, remains an open question. Some partial results ofuniqueness locally in time [10], or for special geometries, like for instance in thin domains [19,14,15] have been obtained,but for general 3D geometries uniqueness may be expected only for a restricted class of more regular solutions (see [6,20]where only homogeneous Dirichlet boundary conditions and constant viscosity are considered).

In this paperwe consider a non constant viscositywhich depends on the temperature T and involved boundary conditionswhich are important in engineering applications, namely Tresca’s friction law on Γ0 combined with a time-dependentnonhomogeneous Dirichlet condition on ΓL (see (1.5)–(1.6) and (1.4)).

Acknowledgements

The authorswould like to thank verymuch the anonymous referees for their helpful commentswhich allowed to improveour paper.

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