Nonlocal Cahn-Hilliard-Navier-Stokes systems with singular potentials

0.32 MB·

by  Sergio Frigeri

NONLOCAL CAHN-HILLIARD-NAVIER-STOKES SYSTEMS
WITH SINGULAR POTENTIALS
Sergio Frigeri
Dipartimento di Matematica F. Enriques
2
1 Universit`a degli Studi di Milano
0
2
Milano I-20133, Italy

n
a [email protected]
J

0
3
Maurizio Grasselli

]
P
Dipartimento di Matematica F. Brioschi
A
h. Politecnico di Milano
t
a
Milano I-20133, Italy
m
[ [email protected]

1
v January 31, 2012
3
0
3
6
. Abstract
1
0 Here we consider a Cahn-Hilliard-Navier-Stokes system characterized by a nonlocal
2
1 Cahn-Hilliard equation with a singular (e.g., logarithmic) potential. This system
:
v originates from a diffuse interface model for incompressible isothermal mixtures of
i
X two immiscible fluids. We have already analyzed the case of smooth potentials with
r arbitrary polynomial growth. Here, taking advantage of the previous results, we
a
study this more challenging (and physically relevant) case. We first establish the
existence of a global weak solution with no-slip and no-flux boundary conditions.
Then we prove the existence of the global attractor for the 2D generalized semiflow
(in the sense of J.M. Ball). We recall that uniqueness is still an open issue even
in 2D. We also obtain, as byproduct, the existence of a connected global attractor
for the (convective) nonlocal Cahn-Hilliard equation. Finally, in the 3D case, we
establish the existence of a trajectory attractor (in the sense of V.V. Chepyzhov
and M.I. Vishik).
Keywords: Navier-Stokes equations, nonlocal Cahn-Hilliard equations, singular
1
potentials, incompressible binary fluids, global attractors, trajectory attractors.
AMS Subject Classification 2010: 35Q30, 37L30, 45K05, 76T99.
1 Introduction
In [12] we have introduced and analyzed an evolution system which consists of the Navier-
Stokesequationsforthefluidvelocityusuitablycoupledwithanon-localconvectiveCahn-
Hilliard equation for the order parameter ϕ on a given (smooth) bounded domain Ω Rd,
⊂
d = 2,3. This system derives from a diffuse interface model which describes the evolution
of an incompressible mixture of two immiscible fluids (see, e.g., [20, 21, 22, 23, 25] and
references therein). We suppose that the temperature variations are negligible and the
density is constant and equal to one. Thus u represents an average velocity and ϕ the
relative concentration of one fluid (or the difference of the two concentrations). Then the
nonlocal Cahn-Hilliard-Navier-Stokes system reads as follows
ϕ +u ϕ = ∆µ, (1.1)
t
·∇
u div(2ν(ϕ)Du)+(u )u+ π = µ ϕ+h, (1.2)
t
− ·∇ ∇ ∇
µ = aϕ J ϕ+F′(ϕ), (1.3)
− ∗
div(u) = 0, (1.4)
in Ω (0,+ ). We endow the system with the boundary and initial conditions
× ∞
∂µ
= 0, u = 0, on ∂Ω, (1.5)
∂n
u(0) = u , ϕ(0) = ϕ , in Ω, (1.6)
0 0
where n is the unit outward normal to ∂Ω. Here ν is the viscosity, π the pressure, h
denotes an external force acting on the fluid mixture, J : Rd R is a suitable interaction
→
kernel, a is a coefficient depending on J (see section below for the related assumptions),
F is the configuration potential which accounts for the presence of two phases.
Here we prove the existence ofa global weak solutionwhen thedouble-well potential F
is assumed to be singular in ( 1,1), that is, its derivative is unbounded at the endpoints.
−
A typical situation of physical interest is the following (see [8])
θ θ
F(s) = ((1+s)log(1+s)+(1 s)log(1 s)) cs2, (1.7)
2 − − − 2
where θ, θ are the (absolute) temperature and the critical temperature, respectively. If
c
0 < θ < θ then phase separation occurs, otherwise the mixed phase is stable. We recall
c
that the logarithmic terms are related to the entropy of the system.
2
For the existence of a weak solution, we take advantage of our previous analysis for
regular potentials (i.e., defined on the whole R) with polynomially controlled growth of
arbitrary order (see [12]) and we use a suitable approximation procedure inspired by [16].
Then, we extend to potentials like (1.7) the results obtained in [17] for regular potentials.
Such results are concerned with the global longtime behavior of (weak) solutions. More
precisely, in the spirit of [4], we can define a generalized semiflow in 2D and prove that it
possesses a global (strong) attractor by using the energy identity. Then we analyze the
3D case by means of the trajectory approach introduced in [26] and generalized in [9, 10].
In this framework, we show the existence of a trajectory attractor.
We recall that the chemical potential of the corresponding local Cahn-Hilliard-Navier-
Stokes system is given by µ = ∆ϕ+F′(ϕ). Therefore it can be seen as an approximation
−
of the nonlocal one (cf. [12] and references therein). The local system with a singular
potential has been analyzed in [1, 2, 7] (for regular potentials see, e.g., [18, 19, 27, 29] and
references therein). Most of the results known for the Navier-Stokes equations essentially
hold for the coupled (local) system as well. On the contrary, in the nonlocal case, due
to the weaker smoothness of ϕ, proving uniqueness and/or getting higher-order estimates
seem a non-trivial task even in dimension two (see [12, 17]).
We conclude by observing that the technique we use in 2D can be easily adapted
to show that the (convective) Cahn-Hilliard equation with a singular potential has a
connected global (strong) attractor (for regular potentials see [17] and references therein,
cf. also [3, 15] for results on the local case).
The plan goes as follows. In the next section, we introduce the weak formulation of
our problem. Then we state the existence theorem whose proof is given in Section 3.
Section 4 is devoted to the global attractor in 2D, while Section 5 is concerned with the
existence of the trajectory attractor.
2 Weak solutions and existence theorem
Let us set H := L2(Ω) and V := H1(Ω). For every f V′ we denote by f the average of
∈
f over Ω, i.e.,
1
f := f,1 .
Ω h i
| |
Here Ω stands for the Lebesgue measure of Ω.
| |
Then we introduce the spaces
V := v V : v = 0 , V′ := f V′ : f = 0 ,
0 { ∈ } 0 { ∈ }
3
and the operator A : V V′, A (V,V′) defined by
→ ∈ L
Au,v := u v u,v V.
h i ∇ ·∇ ∀ ∈
ZΩ
We recall that A maps V onto V′ and the restriction of A to V maps V onto V′ isomor-
0 0 0 0
phically. Let us denote by : V′ V the inverse map defined by
N 0 → 0
A f = f, f V′ and Au = u, u V .
N ∀ ∈ 0 N ∀ ∈ 0
As is well known, for every f V′, f is the unique solution with zero mean value of
∈ 0 N
the Neumann problem
∆u = f, in Ω
−
( ∂u = 0, on ∂Ω.
∂n
Furthermore, the following relations hold
Au, f = f,u , u V, f V′, (2.1)
h N i h i ∀ ∈ ∀ ∈ 0
f, g = g, f = ( f) ( g), f,g V′. (2.2)
h N i h N i ∇ N ·∇ N ∀ ∈ 0
ZΩ
We also consider the standard Hilbert spaces for the Navier-Stokes equations (see, e.g.,
[28])
L2(Ω)d
G := u C∞(Ω)d : div(u) = 0 , V := u H1(Ω)d : div(u) = 0 .
div { ∈ 0 } div { ∈ 0 }
We denote by and ( , ) the norm and the scalar product on both H and G ,
div
k · k · ·
respectively. We recall that V is endowed with the scalar product
div
(u,v) = ( u, v), u,v V .
Vdiv ∇ ∇ ∀ ∈ div
We shall also use the definition of the Stokes operator S with no-slip boundary condition.
More precisely, S : D(S) G G is defined as S := P∆ with domain D(S) =
div div
⊂ → −
H2(Ω)d V , where P : L2(Ω)d G is the Leray projector. Notice that we have
div div
∩ →
(Su,v) = (u,v) = ( u, v), u D(S), v V
Vdiv ∇ ∇ ∀ ∈ ∀ ∈ div
and S−1 : G G is a self-adjoint compact operator in G . Thus, according with
div div div
→
classical spectral theorems, it possesses a sequence λ with 0 < λ λ and
j 1 2
{ } ≤ ≤ ···
λ , and a family w D(S) of eigenfunctions which is orthonormal in G . It is
j j div
→ ∞ { } ⊂
also convenient to recall that the trilinear form b which appears in the weak formulation
of the Navier-Stokes equations is defined as follows
b(u,v,w) = (u )v w, u,v,w V .
div
·∇ · ∀ ∈
ZΩ
4
We suppose that the potential F can be written in the following form
F = F +F ,
1 2
where F C(2+2q)( 1,1), with q a fixed positive integer, and F C2([ 1,1]).
1 2
∈ − ∈ −
We can now list the assumptions on the kernel J, on the viscosity ν, on F , F and on
1 2
the forcing term h.
(A1) J W1,1(Rd), J(x) = J( x), a(x) := J(x y)dy 0, a.e. x Ω.
∈ − − ≥ ∈
ZΩ
(A2) The function ν is locally Lipschitz on R and there exist ν ,ν > 0 such that
1 2
ν ν(s) ν , s R.
1 2
≤ ≤ ∀ ∈
(A3) There exist c > 0 and ǫ > 0 such that
1 0
(2+2q)
F (s) c , s ( 1, 1+ǫ ] [1 ǫ ,1).
1 ≥ 1 ∀ ∈ − − 0 ∪ − 0
(A4) There exists ǫ > 0 such that, for each k = 0,1, ,2+2q and each j = 0,1, ,q,
0
··· ···
(k)
F (s) 0, s [1 ǫ ,1),
1 ≥ ∀ ∈ − 0
(2j+2) (2j+1)
F (s) 0, F (s) 0, s ( 1, 1+ǫ ].
1 ≥ 1 ≤ ∀ ∈ − − 0
(2+2q)
(A5) There exists ǫ > 0 such that F is non-decreasing in [1 ǫ ,1) and non-
0 1 − 0
increasing in ( 1, 1+ǫ ].
0
− −
(A6) There exist α,β R with α+β > min F′′ such that
∈ − [−1,1] 2
′′
F (s) α, s ( 1,1), a(x) β, a.e. x Ω.
1 ≥ ∀ ∈ − ≥ ∈
(A7) lim F′(s) = .
s→±1 1 ±∞
(A8) h L2(0,T;V′ ) for all T > 0.
∈ div
Remark 1. Assumptions (A3)-(A7) are satisfied in the case of the physically relevant
logarithmic double-well potential (1.7) for any fixed positive integer q. In particular,
setting
θ θ
F (s) = ((1+s)log(1+s)+(1 s)log(1 s)), F (s) = cs2,
1 2
2 − − − 2
then (A6) is satisfied if and only if β > θ θ.
c
−
5
Remark 2. The requirement a(x) β a.e x Ω is crucial (see [5, Rem.2.1], cf. also
≥ ∈
[6]). For example, in the case of the double-well smooth potential F(s) = (s2 1)2,
−
which is usually taken as a fairly good smooth approximation of the singular potential,
the existence result in [12] requires the condition a(x) β with β > 4.
≥
The notion of weak solution to problem (1.1)-(1.6) is given by
Definition 1. Let u G , ϕ H with F(ϕ ) L1(Ω) and 0 < T < + be given. A
0 div 0 0
∈ ∈ ∈ ∞
couple [u,ϕ] is a weak solution to (1.1)-(1.6) on [0,T] corresponding to [u ,ϕ ] if
0 0
u, ϕ and µ satisfy
•
u L∞(0,T;G ) L2(0,T;V ), (2.3)
div div
∈ ∩
u L4/3(0,T;V′ ), if d = 3, (2.4)
t ∈ div
u L2(0,T;V′ ), if d = 2, (2.5)
t ∈ div
ϕ L∞(0,T;H) L2(0,T;V), (2.6)
∈ ∩
ϕ L2(0,T;V′), (2.7)
t
∈
µ = aϕ J ϕ+F′(ϕ) L2(0,T;V), (2.8)
− ∗ ∈
and
ϕ L∞(Q), ϕ(x,t) < 1 a.e. (x,t) Q := Ω (0,T); (2.9)
∈ | | ∈ ×
for every ψ V, every v V and for almost any t (0,T) we have
div
• ∈ ∈ ∈
ϕ ,ψ +( µ, ψ) = (u,ϕ ψ), (2.10)
t
h i ∇ ∇ ∇
u ,v +(2ν(ϕ)Du,Dv)+b(u,u,v) = (ϕ µ,v)+ h,v ; (2.11)
t
h i − ∇ h i
the initial conditions u(0) = u , ϕ(0) = ϕ hold.
0 0
•
Theorem 1. Assume that (A1)-(A8) are satisfied for some fixed positive integer q. Let
u G , ϕ L∞(Ω) such that F(ϕ ) L1(Ω). In addition, assume that ϕ < 1.
0 div 0 0 0
∈ ∈ ∈ | |
Then, for every T > 0 there exists a weak solution z := [u,ϕ] to (1.1)-(1.6) on [0,T]
corresponding to [u ,ϕ ] such that ϕ(t) = ϕ for all t [0,T] and
0 0 0
≥
ϕ L∞(0,T;L2+2q(Ω)). (2.12)
∈
Furthermore, setting
1 1
(u(t),ϕ(t)) = u(t) 2 + J(x y)(ϕ(x,t) ϕ(y,t))2dxdy + F(ϕ(t)),
E 2k k 4 − −
ZΩZΩ ZΩ
6
the following energy inequality holds
t t
(u(t),ϕ(t))+ 2 ν(ϕ)Du(τ) 2+ µ(τ) 2 dτ (u(s),ϕ(s))+ h(τ),u(τ) dτ,
E k k k∇ k ≤ E h i
Zs (cid:16) p (cid:17) Zs (2.13)
for all t s and for a.a. s (0, ), including s = 0. If d = 2, the weak solution
≥ ∈ ∞
z := [u,ϕ] satisfies
d
(u,ϕ)+2 ν(ϕ)Du 2 + µ 2 = h,u , (2.14)
dtE k k k∇ k h i
p
i.e., equality holds in (2.13) for every t 0.
≥
Recalling[17, Corollary1, Proposition5], wecanalsodeduceanexistence (andunique-
ness) result for the convective nonlocal Cahn-Hilliard equation with a given velocity field.
Corollary 1. Assume that (A1) and (A3)-(A7) are satisfied for some fixed positive integer
q. Let u L2 ([0, );V L∞(Ω)d) be given and let ϕ L∞(Ω) such that F(ϕ )
∈ loc ∞ div ∩ 0 ∈ 0 ∈
L1(Ω). In addition, suppose that ϕ < 1. Then, for every T > 0, there exists a unique
0
| |
ϕ L2(0,T;V) H1(0,T;V′) which fulfills (2.9) and (2.12), solves (2.10) on [0,T] with
∈ ∩
µ given by (2.8) and initial condition ϕ(0) = ϕ . In addition, for all t 0, we have
0
≥
(ϕ(t),1) = (ϕ ,1) and the following energy identity holds
0
d 1
J(x y)(ϕ(x,t) ϕ(y,t))2dxdy + F(ϕ(t)) + µ 2 = (uϕ, µ).
dt 4 − − k∇ k ∇
(cid:18) ZΩZΩ ZΩ (cid:19)
(2.15)
Remark 3. Note that, thanks to (2.6), (2.8) and (2.13), we have that
F′(ϕ) L2(0,T;V), F(ϕ) L∞(0,T;L1(Ω)), T > 0.
∈ ∈ ∀
Remark 4. The regularity property (2.12) does not follow from (2.9). Indeed, recall that
L∞(0,T;L∞(Ω)) L∞(Q) with strict inclusion.
⊂
3 Proof of Theorem 1
We consider the following approximate problem P : find a weak solution [u ,ϕ ] to
ǫ ǫ ǫ
ϕ′ +u ϕ = ∆µ , (3.1)
ǫ ǫ ·∇ ǫ ǫ
u′ div(ν(ϕ )2Du )+(u )u + π = µ ϕ +h, (3.2)
ǫ − ǫ ǫ ǫ ·∇ ǫ ∇ ǫ ǫ∇ ǫ
µ = aϕ J ϕ +F′(ϕ ), (3.3)
ǫ ǫ − ∗ ǫ ǫ ǫ
div(u ) = 0, (3.4)
ǫ
7
∂µ
ǫ
= 0, u = 0, on ∂Ω, (3.5)
ǫ
∂n
u (0) = u , ϕ (0) = ϕ , in Ω. (3.6)
ǫ 0 ǫ 0
Problem P is obtained from (1.1)-(1.6) by replacing the singular potential F with the
ǫ
smooth potential
F = F +F ,
ǫ 1ǫ 2
where F is defined by
1ǫ
(2+2q)
F (1 ǫ), s 1 ǫ
1 − ≥ −
F(2+2q)(s) = F(2+2q)(s), s 1 ǫ (3.7)
1ǫ  1 | | ≤ −
 F(2+2q)( 1+ǫ), s 1+ǫ
1 − ≤ −

and F (0) = F (0), F′ (0) = F′(0),...F(1+2q)(0) = F(1+2q)(0), while F is a C2(R)-
1ǫ 1 1ǫ 1 1ǫ 1 2
extension of F on R with polynomial growth satisfying
2
F (s) min F 1, F′′(s) min F′′, s R. (3.8)
2 ≥ [−1,1] 2 − 2 ≥ [−1,1] 2 ∀ ∈
The following elementary lemmas are basics to obtain uniform (w.r.t. ǫ) estimates for
a weak solution to the approximate problem.
Lemma 1. Suppose that (A3) and (A4) hold. Then, there exist c ,d > 0, which depend
q q
on q but are independent of ǫ, and ǫ > 0 such that
0
F (s) c s 2+2q d , s R, ǫ (0,ǫ ]. (3.9)
ǫ q q 0
≥ | | − ∀ ∈ ∀ ∈
Proof. By integrating (3.7) we get
2+2q 1F(k)(1 ǫ)[s (1 ǫ)]k, s 1 ǫ
k=0 k! 1 − − − ≥ −
F (s) = F (s), s 1 ǫ (3.10)
1ǫ  P1 | | ≤ −
 2+2q 1F(k)( 1+ǫ)[s ( 1+ǫ)]k, s 1+ǫ.
k=0 k! 1 − − − ≤ −
Due to (A4) we haveP, for ǫ small enough,
1
F (s) F(2+2q)(1 ǫ)[s (1 ǫ)]2+2q, s 1 ǫ,
1ǫ ≥ (2+2q)! 1 − − − ∀ ≥ −
so that, in particular,
1
F (s) F(2+2q)(1 ǫ)(s 1)2+2q, s 1,
1ǫ ≥ (2+2q)! 1 − − ∀ ≥
and (A3) implies that (for ǫ small enough)
F (s) 2c (s 1)2+2q c s2+2q d , s 1,
1ǫ q q q
≥ − ≥ − ∀ ≥
8
where c = c /2(2+2q)! and d is another constant depending only on q. Furthermore,
q 1 q
we have F (s) = F (s) 0 c s2+2q d for 0 s 1 ǫ, provided we choose d c ,
1ǫ 1 q q q q
≥ ≥ − ≤ ≤ − ≥
while for 1 ǫ s 1 we have F 2c [s (1 ǫ)]2+2q 0 c s2+2q d . Summing
1ǫ q q q
− ≤ ≤ ≥ − − ≥ ≥ −
up, we deduce that there exists ǫ > 0 such that F (s) c s2+2q d , for all s 0 and
0 1ǫ q q
≥ − ≥
for all ǫ (0,ǫ ]. By using (3.8) we also get (3.9) for s 0. Similarly we obtain (3.9) for
0
∈ ≥
s 0.
≤
Lemma 2. Suppose (A4) and (A6) hold. Then, setting c := α + β +min F′′ > 0,
0 [−1,1] 2
there exists ǫ > 0 such that
1
F′′(s)+a(x) c , s R, a.e. x Ω, ǫ (0,ǫ ]. (3.11)
ǫ ≥ 0 ∀ ∈ ∈ ∀ ∈ 1
Proof. From (3.10) we have
2q 1F(k+2)(1 ǫ)[s (1 ǫ)]k, s 1 ǫ
k=0 k! 1 − − − ≥ −
F′′(s) = F′′(s), s 1 ǫ (3.12)
1ǫ  P1 | | ≤ −
 2q 1F(k)( 1+ǫ)[s ( 1+ǫ)]k, s 1+ǫ.
k=0 k! 1 − − − ≤ −
Assumption (A4)imPplies that for ǫ small enough F′′(s) F′′(1 ǫ) for s 1 ǫ and
1ǫ ≥ 1 − ≥ −
F′′(s) F′′( 1 + ǫ) for s 1 + ǫ. Since F′′(s) = F′′(s) for s 1 ǫ, (A6) implies
1ǫ ≥ 1 − ≤ − 1ǫ 1 | | ≤ −
that there exists ǫ > 0 such that
1
F′′(s) α, s R, ǫ (0,ǫ ]. (3.13)
1ǫ ≥ ∀ ∈ ∀ ∈ 1
This estimate together with (3.8) and (A6) imply (3.11).
Due to the existence result proved in [12], for every T > 0, Problem P admits a weak
ǫ
solution z := [u ,ϕ ] such that
ǫ ǫ ǫ
u L∞(0,T;G ) L2(0,T;V ), (3.14)
ǫ div div
∈ ∩
u′ L4/3(0,T;V′ ), if d = 3, (3.15)
ǫ ∈ div
u′ L2(0,T;V′ ), if d = 2, (3.16)
ǫ ∈ div
ϕ L∞(0,T;L2+2q(Ω)) L2(0,T;V), (3.17)
ǫ
∈ ∩
ϕ′ L2(0,T;V′), (3.18)
ǫ ∈
µ L2(0,T;V). (3.19)
ǫ
∈
Indeed, it is immediate to check that all the assumptions of [12, Theorem 1] and of
[12, Corollary 1] are satisfied for Problem P . In particular, we use Lemma 1, Lemma 2
ǫ
and the fact that, due to the definition of F and to the polynomial growth assumption
1ǫ
on F , assumption (H5) of [12, Theorem 1] is trivially satisfied for each ǫ > 0 (with some
2
constants depending on ǫ).
9
Furthermore, according to [12, Theorem 1] and using (A2), the approximate solution
z := [u ,ϕ ] satisfies the following energy inequality
ǫ ǫ ǫ
1 1
u (t) 2 + J(x y)(ϕ (x,t) ϕ (y,t))2dxdy + F (ϕ (t))
ǫ ǫ ǫ ǫ ǫ
2k k 4 − −
ZΩZΩ ZΩ
t 1 1
+ (ν u 2 + µ 2)dτ u 2 + J(x y)(ϕ (x) ϕ (y))2dxdy
1 ǫ ǫ 0 0 0
k∇ k k∇ k ≤ 2k k 4 − −
Z0 ZΩZΩ
t
+ F (ϕ )+ h,u dτ, t [0,T]. (3.20)
ǫ 0 ǫ
h i ∀ ∈
ZΩ Z0
From (A5) it is easy to see (cf. (3.33) and (3.34) below) that there exists ǫ > 0 such that
1
F (s) F (s), s ( 1,1), ǫ (0,ǫ ]. (3.21)
1ǫ 1 1
≤ ∀ ∈ − ∀ ∈
Therefore, using the assumptions onϕ , u and Lemma 1, from(3.20) we get the following
0 0
estimates
kuǫkL∞(0,T;Gdiv)∩L2(0,T;Vdiv) ≤ c, (3.22)
ϕǫ L∞(0,T;L2+2q(Ω)) c, (3.23)
k k ≤
µ c. (3.24)
ǫ L2(0,T;H)
k∇ k ≤
Henceforth c will denote a positive constant which depends on the initial data, but is
independent of ǫ.
Wethentake thegradient of (3.3)andmultiply theresulting identity by ϕ inL2(Ω).
ǫ
∇
Arguing as in [12], we get
c2
µ 2 0 ϕ 2 k ϕ 2,
ǫ ǫ ǫ
k∇ k ≥ 4 k∇ k − k k
with k = 2 J 2 . This last estimate together with (3.23) and (3.24) yield
k∇ kL1
ϕ c. (3.25)
ǫ L2(0,T;V)
k k ≤
As far as the bounds on the time derivatives u′ and ϕ′ are concerned, on account of
{ ǫ} { ǫ}
(3.1) and (3.2), arguing by comparison as in [12] one gets
kϕ′ǫkL2(0,T;V′) ≤ c, (3.26)
ku′ǫkL2(0,T;Vd′iv) ≤ c, d = 2 (3.27)
ku′ǫkL4/3(0,T;Vd′iv) ≤ c, d = 3. (3.28)
In order to obtain an estimate for µ we need to control the sequence of averages µ .
ǫ ǫ
{ } { }
To this aim observe that equation (3.1) can be written in abstract form as follows
ϕ′ +u ϕ = Aµ in V′. (3.29)
ǫ ǫ ·∇ ǫ − ǫ
10