Nonsimple isotropic incompressible linear fluids surrounding one-dimensional structures

Acta Mech 217, 191–204 (2011)DOI 10.1007/s00707-010-0387-5

Giulio G. Giusteri · Alfredo Marzocchi · Alessandro Musesti

Nonsimple isotropic incompressible linear fluidssurrounding one-dimensional structures

Received: 16 April 2010 / Revised: 10 September 2010 / Published online: 20 October 2010© Springer-Verlag 2010

Abstract We introduce a model of fluid which has four main features: it readily emerges by a general continuummechanical framework; it is a generalization maintaining most of the physical features of incompressible New-tonian fluids; it can model adherence interactions with one-dimensional structures surrounded by the fluid; theassociated initial boundary-value problem is well-posed on three-dimensional domains.

1 Introduction

In the last decades, it has become a prominent question how to manage the interplay between parts of acomposite system which are characterized by very different typical sizes, especially within the frameworkof numerical approximation. More and more biological and environmental applications require the treatmentof “small” objects immersed in a wider context, giving rise to the issue of describing them with a suitabledegree of accuracy, looking for a delicate balance between computability and reliability of the approximation.When the main character of the play is the wider context, it is often unnecessary to specify all the physicsof the smaller structures: it is enough to capture the effects of their presence on the larger system. Of course,there are many examples of the opposite situation, but even when the physics of the small-size structuresis important, their geometry can be approximated by means of lower dimensional structures, still obtaininga good description of the whole.

In this paper, we introduce a model of fluid which addresses four main features:

– it readily emerges by a general continuum mechanical framework;– it is a generalization maintaining most of the physical features of incompressible Newtonian fluids;– it can model adherence interactions with one-dimensional structures;– the initial boundary-value problem enjoys well-posedness on three-dimensional domains.

The general continuum mechanical framework we refer to is that of second-gradient continuous media (whichwe prefer to name second-order media for a reason clarified after Definition 1) introduced by Fried and Gur-tin [4] within a virtual power format; we apply their theory to the most general incompressible linear isotropic

G. G. GiusteriDipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca,Via Cozzi, 53, 20125 Milan, ItalyE-mail: [email protected]

A. Marzocchi · A. Musesti (B)Dipartimento di Matematica e Fisica “N. Tartaglia”, Università Cattolica del Sacro Cuore,Via dei Musei, 41, 25121 Brescia, ItalyE-mail: [email protected]

A. MarzocchiE-mail: [email protected]

192 G. G. Giusteri et al.

second-order fluid, whose constitutive prescriptions have been set down by Musesti [12], establishing alsonecessary and sufficient conditions for the viscous dissipation to meet thermodynamical feasibility.

Since the constitutive assumptions, encoding the physics, are akin to the Newtonian ones, it can be expecteda quasi-Newtonian behavior, and it actually appears, as we will highlight in Sect. 3; moreover, the introductionof terms involving higher-order derivatives of the velocity field, which is the mark of second-order theories,can suggest the interpretation of our model as a singular perturbation of a Newtonian fluid. This kind of per-turbation induces regularizing effects: on the one hand, the well-posedness theory becomes an application ofwell-known results in PDE’s theory and functional analysis, and on the other hand, the regularity producedis the mathematical key for the management of a nonstandard fluid–structure interaction. In fact we will dealwith continuous velocity fields in R

3, which in general is not the case for Newtonian fluids, and this will enableus to impose the adherence of our fluid to one-dimensional structures surrounded by the fluid.

2 The model

A general way to set down a thermomechanical model for continuous media is to apply the principle of virtualpowers as an axiom [5,11]. As pointed out by Podio-Guidugli in [14], local thermomechanical balances canbe deduced from the balance of properly defined internal and external powers. Since our aim is to apply thevirtual powers framework to Fluid Mechanics in three-dimensional space, we will assume an Eulerian pointof view, interpreting test functions as velocity fields in the definition of the power; moreover, we will restrictour attention to a pure mechanical theory.

Within this context, we introduce the relevant notion of power of order k.

Definition 1 Given Ω ⊆ R3, a suitable1 collection M of subsets of Ω and a vector space V of virtual velocities

v : Ω → R3, we call power of order k a functional

P :⎧⎨

⎩

M × V → R

(M, v) �→k∑

j=0

∫

MA( j)(x) · ∇ j v(x)

(1)

where A( j)(x) belongs to the same linear space as ∇ j v(x).

In order to endow the above definition at least with a distributional meaning, it is customary to choose a spaceV containing C∞

0 (Ω; R3).

We name a fluid of the second order after the order of the power Pint(M, v) expended by internal stresseson a virtual velocity v. In the meanwhile, one can consider the power Pext(M, v) expended by external forcesand inertial ones. Physical situations are characterized by the balance of such power expenditures, and thenthe following fundamental principle provides the equation to be solved in order to describe the dynamics ofthe system.

Principle of virtual powers The motion is feasible if and only if for every M ∈ M and every virtual velocityv we have

Pint(M, v) = Pext(M, v), (2)

for any instant in a time interval.

We will now turn to the task of specifying a model of homogeneous fluid by means of constitutive pre-scriptions; however, we want to emphasize that also the prescription on the order of the power is a matter ofchoice of the model, and it does not enter the general assumptions of Continuum Mechanics.

1 We refer to [1,2,6,13] for some important contributions and discussions about the mathematical translation of this suitable.

Nonsimple linear fluids surrounding one-dimensional structures 193

2.1 Mechanical constraints

The descriptor of the state at any instant t in the time interval [0, T ] ⊂ R is the Eulerian velocity field u(t, x),and the incompressibility condition, together with the assumption of homogeneity, allows us to set the massdensity ρ = 1 identically, giving the first constraint on the velocity:

∀t ∈ [0, T ] : div u = 0. (3)

Further prescriptions are related to internal and external powers.A second-order internal power is required to satisfy Galilean invariance. Then, its general form becomes

Pint(M, v) =∫

M

T · ∇v +∫

M

G · ∇∇v,

where T is a symmetric tensor field of order 2 and G a tensor field of order 3. Linearity and isotropy (i.e.covariance under the full orthogonal group on R

3) of the fluid are encoded in the dependence of the tensorfields T and G on u. It is well-known that within incompressible theories

Ti j = μ(ui, j + u j,i ) − p δi j ;besides, in [12, Theorem 1.1] it has been shown that

Gi jk = η1ui, jk + η2(u j,ki + uk,i j − ui,ssδ jk)

+ η3(u j,ssδki + uk,ssδi j − 4ui,ssδ jk) − pkδi j ,

where μ, η1, η2, η3 ∈ R and δi j is the usual Kronecker symbol. The fields p and p, respectively a scalar anda vector one, enter the definition of the pressure

P := p − div p,

whose role in incompressible theories reduces to that of a Lagrange multiplier of the constraint (3).Defining the symmetric part of an m-tensor X as

Sym X := 1

m!∑

σ

Xσ(i1,...,im),

where σ runs over the group of permutations of m elements, and setting I = (δi j ), the previous relations canbe written in intrinsic notation as

T = 2μ Sym ∇u − p I,G = (η1 − η2)∇∇u + 3η2 Sym ∇∇u

− (η2 + 5η3)�u ⊗ I + 3η3 Sym (�u ⊗ I) − I ⊗ p.

Following these definitions and imposing also on the virtual velocities the constraint (3), we can write theinternal power for a linear isotropic incompressible fluid as

Pint(M, v) = 2μ

∫

M

Sym ∇u · ∇v + (η1 − η2)

∫

M

∇∇u · ∇∇v

+ 3η2

∫

M

Sym ∇∇u · ∇∇v − (η2 + 4η3)

∫

M

�u · �v.

As clearly explained by Podio-Guidugli and Vianello [15], the internal second-order power can be charac-terized in terms of:


– a volume interaction field b̂;– traction and hypertraction fields t̂ and m̂ as surface interactions on the smooth part of ∂ M ;– the field f̂c as concentrated interaction2 on the nonsmooth part of the boundary.

This equivalent representation of the internal power plays a basic role when we model the action of the envi-ronment on our system: it is important to make the external world act in a way that can be experienced bythe system. Such requirement can be met by defining the external power expended on the whole region Ω inanalogy to the internal one. Using the interaction-fields representation, we can introduce:

– an external volume interaction field b;– external traction and hypertraction fields t and m as surface interactions on appropriate smooth parts of

∂Ω;– the field fc as concentrated interaction on nonsmooth parts of the boundary, if any.

As to the external volume interaction b, it is useful to single out the conservative contribution, which playsthe role of a pressure term in an incompressible theory. Hence, we write

b = d + ∇ f,

where f is a scalar field and d includes all nonconservative contributions to the volume interactions of themedium. Statements about the surface interactions, concerning boundary conditions, will appear in Sect. 3.2,while here we do not employ concentrated interactions at the boundary of the whole body Ω . ApplyingD’Alembert’s principle, we also include in the external power the inertial term

−∫

M

ρ u̇ · v := −∫

M

ρ

(∂u∂t

+ (u · ∇)u)

· v,

which makes the problem a nonlinear one.

2.2 Thermodynamical constraints

By thermodynamical considerations, we need the instantaneous dissipation to be nonnegative for any flow;hence, we require the dissipation inequality

T · ∇v + G · ∇∇v ≥ 0 (4)

to be satisfied for every velocity v. This inequality specialized for our model reads

2μ|Sym ∇v|2 + (η1 − η2)|∇∇v|2 + 3η2|Sym ∇∇v|2 − (η2 + 4η3)|�v|2 ≥ 0

for every virtual velocity v.Since the first- and second-order derivatives of v can be independently set equal to zero, the dissipation

inequality will be satisfied if and only if μ ≥ 0 and

Γ := ∇∇v · G[∇∇v] = η1vi, jkvi, jk + η2(2vk,i j vi, jk − vi,rr vi,ss) − 4η3vi,rr vi,ss ≥ 0

for every virtual velocity v. This last requirement is equivalent to the following conditions on the coefficientsη1, η2 and η3:

Proposition 1 We have Γ ≥ 0 for every virtual velocity v if and only if 3

η1 + 2η2 ≥ 0, η1 − η2 ≥ 0, η1 − η2 − 6η3 − 2√

η22 + 4η2η3 + 9η2

3 ≥ 0. (5)

2 A representation theorem for edge-force densities, stated within the general context of Geometric Measure Theory, can befound in [2, Sec. 4.2].

3 With the choice η3 = 0 one finds η1 + 2η2 ≥ 0 and η1 − 3η2 ≥ 0. The reader should compare such inequalities with [4, Eq.(75)], which are only sufficient conditions.


Proof Let us identify the 18 independent components of ∇∇v with an element x ∈ R18 according to the

following table:

x1 = v1,11 x2 = v1,22 x3 = v1,33 x4 = v2,12 x5 = v3,13x6 = v2,22 x7 = v2,33 x8 = v2,11 x9 = v3,23 x10 = v1,12x11 = v3,33 x12 = v3,11 x13 = v3,22 x14 = v1,13 x15 = v2,23x16 = v1,23 x17 = v2,13 x18 = v3,12.

Then, we can write

Γ = x · (η1A + B)x, (6)

where A = diag(A5, A5, A5, A3), B = diag(B5, B5, B5, B3) and

A5 = diag(1, 1, 1, 2, 2), A3 = diag(2, 2, 2),

B5 =

⎡

⎢⎢⎢⎢⎢⎢⎣

η2 − 4η3 −η2 − 4η3 −η2 − 4η3 0 0

−η2 − 4η3 −η2 − 4η3 −η2 − 4η3 2η2 0

−η2 − 4η3 −η2 − 4η3 −η2 − 4η3 0 2η2

0 2η2 0 2η2 0

0 0 2η2 0 2η2

⎤

⎥⎥⎥⎥⎥⎥⎦

, B3 = η2

⎡

⎢⎣

0 1 1

1 0 1

1 1 0

⎤

⎥⎦ .

The quadratic form (6) is positive definite if and only if its eigenvalues are all positive. Since A is positivedefinite, this is tantamount to say that the eigenvalues of η1I + A−1B are positive definite.

Since A−1B = diag(A−15 B5, A−1

5 B5, A−15 B5, A−1

3 B3) and

A−15 B5 =

⎡

⎢⎢⎢⎢⎢⎢⎣

η2 − 4η3 −η2 − 4η3 −η2 − 4η3 0 0

−η2 − 4η3 −η2 − 4η3 −η2 − 4η3 2η2 0

−η2 − 4η3 −η2 − 4η3 −η2 − 4η3 0 2η2

0 η2 0 η2 0

0 0 η2 0 η2

⎤

⎥⎥⎥⎥⎥⎥⎦

,

A−13 B3 = η2

⎡

⎢⎣

0 1/2 1/2

1/2 0 1/2

1/2 1/2 0

⎤

⎥⎦ ,

a straightforward calculation shows that the eigenvalues of A−1B are

λ1,2 = −η2 − 6η3 ± 2√

η22 + 4η2η3 + 9η2

3, λ3,4 = ±η2, λ5 = 2η2, λ6 = −η2

2.

Hence, Γ ≥ 0 for every velocity field if and only if η1 + λmin ≥ 0, where λmin is the minimal eigenvalue.Since

⎧⎪⎨

⎪⎩

λmin = −η2 − 6η3 − 2√

η22 + 4η2η3 + 9η2

3 if η2 + 4η3 ≥ 0λmin = −η2 if η2 + 4η3 ≤ 0 and η2 ≥ 0λmin = 2η2 if η2 + 4η3 ≤ 0 and η2 ≤ 0,

one has the global conditions (5). �

3 Nets in a pipe

We are interested in the description of the flow in a pipe, whose boundary could reasonably satisfy quitestrong regularity assumptions, in which some objects represented by one-dimensional net-like structures areimmersed. Due to these insertions, the global regularity of the domain weakens and we will need to be careful in


proving the well-posedness of our problem. Before entering the mathematical details, we want to motivate witha simple example the attribute quasi-Newtonian for our model and why one can expect it to allow interactionswith lower dimensional structures.

Consider the following situation: an incompressible fluid moves in the cavity between two coaxial cylindersof radii R1 < R2 and the motion is produced imposing an uniform pressure gradient Cez (C ∈ R) on the z = 0circular section, being the axis of the cylinders along the z direction ez . Looking for cylindrically symmetricstationary solutions u(r)ez , where r2 = x2 + y2, in the case of Newtonian liquids with viscosity μ and theusual adherence condition, one finds

u(r) = C

4μ

(

r2 − R22 − R2

1 − R22

log(R1/R2)log(r/R2)

)

for the axial component of the velocity field. It is clear that we can take the inner radius R1 arbitrarily small,but not zero, otherwise it would be impossible to meet the boundary conditions with a continuous solution.Moreover, it would be meaningless to assign the value of the velocity field on a single line (the degenerateinner cylinder) in the usual context of the Sobolev space H1(R3).

With a second-order fluid the analogous problem gives the family of solutions

u(r) = C

4μr2 + α1 + α2 I0

( r

L

)+ α3 log r + α4 K0

( r

L

),

where αi , i = 1, . . . , 4, are constants fixed by the boundary conditions, L is a parameter arising from thehigher-order terms and I0, K0 are Bessel functions of imaginary argument [8, Sec. 5.7]. If we now set R1 = 0,the solution remains bounded provided that α3 = α4, since

limr→0

(log r + K0

( r

L

))< +∞;

moreover, one can still meet the boundary conditions.This example suggests that it is actually possible, by means of second-order theories, to manage adherence

to parts of the boundary with zero surface measure. Moreover, from a viscometric point of view, an explicitcalculation shows that for a linear second-order fluid the shear rate depends linearly on the shear stress, as inthe Newtonian case: hence we can say that our fluid is a quasi-Newtonian one.

3.1 Conditions on the domain

The conditions for an open set to be a suitable domain for our problem arise from three issues:

– to model pipes containing net-like structures;– to apply techniques from the theory of Sobolev spaces;– to model adherence conditions on the structures.

A good choice in order to fulfill all the requirements is the following: we consider a bounded Lipschitz domainΩ representing the pipe, while the one-dimensional structure immersed in it is represented by a closed subsetΛ of Ω , satisfying 0 < H1(Λ) < +∞ (where Hn is the n-dimensional Hausdorff measure)4.

According to these settings, we will work with a domain Ω for which Sobolev’s embeddings hold and alsoKorn’s inequality can be established [7, Sec. 5.6]. We will see below how to impose the adherence on Λ.

3.2 Boundary conditions

Once selected a suitable class of domains, we have to discuss boundary conditions. As anticipated, we areprimarily concerned with modeling adherence to immersed structures, but we also impose no-slip conditionson a part of the boundary denoted by SW , representing the wall of the pipe, for which it is reasonable to requireH2(SW ) > 0. We partition the remaining part of the boundary into two regions, SD and SF , on which differentconditions will be imposed.

4 Indeed our results hold for a generic closed subset Λ ⊂ Ω with H3(Λ) = 0.


The general statement of adherence condition for Newtonian fluids would only require the velocity fieldu on the boundary to be equal to the velocity of the boundary, being the traction Tn (where n is the unit outernormal to the boundary) computed once the problem is solved. Within second-order theories, we are expectedto say something also on derivatives of u at the boundary, hence we introduce a new requirement: in order toforce the value of the hypertraction m̂ := (Gn)n on SW , we add the term

∫

SW

m · ∂v∂n

to the external power; this corresponds to the setting m̂ = m on SW , and different kinds of adherence couldbe modeled by prescriptions of the field m.

We assume vanishing hypertraction and adherence to fixed boundaries, i.e. m = 0 and u = 0 on SW ; weimpose on SD a constant and uniform normal pressure gradient qn, q ∈ R, which drives the fluid flow, addingthe term

∫

SD

qn · v

to the external power; we prescribe no external action on the remaining part of the boundary, so that m = 0and t = 0 on SF .

The adherence to the one-dimensional still structures does not appear within the boundary conditions: itwill be encoded in the definition of an appropriate functional space X , which is given at the beginning ofSect. 4. With these boundary conditions, we do not have to require additional regularity, neither for the domainΩ nor for Λ.

4 Well-posedness

We now want to investigate existence and uniqueness of solutions for both the stationary and the evolutionarymotion of a second-order incompressible fluid on a bounded Lipschitz domain Ω , in which a one-dimensionalstructure Λ is immersed. The assumption of the balance principle (2) in integral form leads directly to aninterpretation of the functions as defined up to negligible sets (with zero Lebesgue measure). We take the setof test functions (virtual velocities) as a linear space with the useful topological structure of a Hilbert space,endowing the principle of virtual powers with a natural interpretation as equality of linear forms.

We construct a Hilbert subspace X of the Sobolev space H2(Ω; R3) in the following way: we set

V := {v|Ω : v ∈ C∞

0 (R3; R3), div v = 0

}

and denote with H and H2d the completions of V in L2(Ω; R

3) and H2(Ω; R3) respectively. Since H2(Ω; R

3)

is continuously embedded in C0(Ω; R3), we can define in an obvious way the closed subspace

Y := {v ∈ H2(Ω; R

3) : v = 0 on SW ∪ Λ};

finally we set X := H2d ∩ Y endowed with the H2(Ω; R

3) norm

‖v‖2X :=

∫

Ω

|v|2 +∫

Ω

|∇v|2 +∫

Ω

|∇∇v|2,

that encodes the natural regularity requested by the problem.It is apparent that velocity fields belonging to the space X vanish on SW ∪ Λ, while their derivatives in

general do not; hence, we can model adherence to walls and to one-dimensional immersed structures in a wayconsistent with the boundary conditions of Sect. 3.2. It remains to specify the role of the time variable t , whichclearly enters the problem via the time derivative of u. As a first step, we could take u ∈ L2([0, T ]; X), so thatu(t, ·) ∈ X for almost every t ∈ [0, T ]; but we will see that if u is a solution of our problem, then

u ∈ L2([0, T ]; X) ∩ C0([0, T ]; H) ∩ H1([0, T ]; X ′) =: X.

Given the representations of internal and external powers introduced in the previous sections, the weakform of the equation of motion corresponds to the following statement:


Problem 1 Find u ∈ X such that for every t ∈ [0, T ]

2μ

∫

Ω

Sym ∇u · ∇v + (η1 − η2)

∫

Ω

∇∇u · ∇∇v

+ 3η2

∫

Ω

Sym ∇∇u · ∇∇v − (η2 + 4η3)

∫

Ω

�u · �v

+∫

Ω

(∂u∂t

+ (u · ∇)u)

· v =∫

SD

qn · v +∫

Ω

d · v, (7)

for every v ∈ X.

When stating the evolutionary problem, we will add an initial condition, while to obtain the stationary problemit is enough to set the time derivative of u equal to zero and to change X into X in the previous statement.

4.1 Compactness of the Navier operator

In Eq. (7), the only nonlinearity is the convective term (u · ∇)u; clearly, it is the source of the main difficultiesin solving our problem. We will work it out via a topological method in which compactness is the key tool;therefore, we introduce some considerations about the compactness properties of that term.

Consider the bilinear function

F :{

H2 × H2 → L2

(u, v) �→ (u · ∇)v ;

by Hölder’s inequality, since H2(Ω; R3) is embedded in L∞(Ω; R

3), we have5

‖F(u, v)‖L2 = ‖(u · ∇)v‖L2 ≤ ‖u‖L∞ ‖∇v‖L2 ≤ c0 ‖u‖H2 ‖v‖H2 (8)

for any u, v ∈ H2(Ω; R3); hence F is continuous.

Theorem 1 The Navier operator K0(u) := F(u, u) is compact from X to X ′.

Proof As already noticed, the bilinear function F is continuous and so is K0, by composition with the function{ u �→ (u, u) }. Moreover, by virtue of (8), it is bounded on bounded subsets of X .

Since X ⊆ H2(Ω; R3) is compactly embedded in L2(Ω; R

3), we can identify L2(Ω; R3) with its dual

space and apply Schauder’s theorem to obtain L2(Ω; R3) compactly embedded in X ′. �

When considering the evolutionary problem, we will need K0 to be a compact operator from L2([0, T ]; X)into L2([0, T ]; X ′). We exploit the following lemma proved in [10, Chap. 1, Sec. 5.2].

Lemma 1 Given three Banach spaces B0 ⊂ B ⊂ B1 with B0 and B1 reflexive and B0 compactly embeddedin B, and for T ∈ (0,+∞) and p0, p1 ∈ (1,+∞) fixed, set

W :={

v : v ∈ L p0([0, T ]; B0),∂v∂t

∈ L p1([0, T ]; B1)

}

.

Then, W is a Banach space compactly embedded in L p0([0, T ]; B).

Theorem 2 The Navier operator K0 is compact from the space

X = L2([0, T ]; X) ∩ C0([0, T ]; H) ∩ H1([0, T ]; X ′)

into L2([0, T ]; X ′).5 Throughout the whole section ci , i ∈ N, denotes a positive constant depending only upon the geometry of Ω .


Proof Since X is compactly embedded in L3(Ω; R3) and so is L

32 (Ω; R

3) into X ′, we can apply Lemma 1with X ⊂ L3(Ω; R

3) ⊂ X ′ and p0 = p1 = 2. It remains to show that K0, as an operator with range in

L2([0, T ]; L32 (Ω; R

3)), is bounded on bounded subsets of its domain. The estimate

T∫

0

‖F(u, v)‖2

L32

dt ≤T∫

0

‖∇v‖2L6 ‖u‖2

L2 ≤ c21

T∫

0

‖v‖2H2 ‖u‖2

L2

gives the needed property since u belongs to L∞([0, T ]; H). �

4.2 The stationary problem

We will first solve the stationary version of Problem 1, in which time dependence is suppressed. Stationarysolutions represent an important class in Fluid Mechanics for many applications; moreover, as we will seelater, the treatment of the evolutionary problem goes along the same way of the stationary one.

Let us define the bilinear form a : X × X → R as follows:

a(u, v) := 2μ

∫

Ω

Sym ∇u · ∇v + (η1 − η2)

∫

Ω

∇∇u · ∇∇v

+ 3η2

∫

Ω

Sym ∇∇u · ∇∇v − (η2 + 4η3)

∫

Ω

�u · �v.

In view of the dissipation inequality (4) and Proposition 1, we will assume a slightly stronger hypothesis onthe sign of the coefficients.

Proposition 2 Provided that

μ > 0, η1 + 2η2 > 0, η1 − η2 > 0, η1 − η2 − 6η3 − 2√

η22 + 4η2η3 + 9η2

3 > 0,

the bilinear form a(u, v) is continuous and coercive on X.

Proof The continuity of a is apparent. With the notation of Proposition 1, we have:

a(u, u) = 2μ

∫

Ω

|Sym ∇u|2 +∫

Ω

Γ ≥ 2μ ‖Sym ∇u‖2L2 + (η1 + λmin) ‖∇∇u‖2

L2 ;

by an application of Korn’s inequality [7, Lemma 6.2], there exists κ > 0 such that

a(u, u) ≥ κ(‖u‖2L2 + ‖∇u‖2

L2) + (η1 + λmin) ‖∇∇u‖2L2 .

Setting

ν := min { κ, η1 + λmin } > 0,

we have a(u, u) ≥ ν ‖u‖2X . �

Consider now the trilinear form b : H2 × H2 × H2 → R given by

b(u, v, w) :=∫

Ω

F(u, v) · w =∫

Ω

(u · ∇)v · w,

which is indeed continuous since

|b(u, v, w)| ≤ ‖F(u, v)‖L2 ‖w‖L2 ≤ c0 ‖u‖H2 ‖v‖H2 ‖w‖H2

for every u, v, w ∈ H2(Ω; R3).


Lemma 2 For every u ∈ X we have b(u, u, u) = 0 .

Proof By standard formulae in tensor calculus we get the assertion for u ∈ V and we can extend it by a densityargument. �

In the course of the proof of the existence result for solutions of the stationary problem, we will apply alsothe following theorem [3, Corollary 8.1].

Theorem (Fixed Point) Let X be a Banach space and Φ : X → X a compact operator. Then either Φ(u) = uhas a solution, or the set

S = {u ∈ X : Φ(u) = λu for some λ > 1}is unbounded.

Theorem 3 There exists u ∈ X such that for every v ∈ X,

a(u, v) + b(u, u, v) = 〈ϕ, v〉, (9)

where ϕ ∈ X ′ is the linear form defined by

〈ϕ, v〉 =∫

SD

qn · v +∫

Ω

d · v.

Proof By the Lax-Milgram theorem, the function L : X → X ′ defined by

∀v ∈ X : 〈L(u), v〉 = a(u, v)

is a homeomorphism. We have

L(u) + K0(u) = ϕ in X ′ (10)

and then

u = L−1 (ϕ − K0(u)) =: Φ(u).

Assume that u ∈ X is a solution of (10); it means that for every v ∈ X

〈L(u), v〉 + 〈K0(u), v〉 = 〈ϕ, v〉.Taking v = u and applying Lemma 2, we can write

ν ‖u‖2X ≤ a(u, u) ≤ |b(u, u, u)| + |〈ϕ, u〉| ≤ ‖ϕ‖X ′ ‖u‖X ,

from which we have the a priori estimate

‖u‖X ≤ 1

ν‖ϕ‖X ′ =: R < +∞. (11)

Take now λ > 1 and assume that Φ(u) = λu; it means that

〈L(λu), u〉 + 〈K0(u), u〉 = 〈ϕ, u〉and, following the argument leading to (11), we obtain ‖u‖X ≤ λ−1 R < R .

Hence the set S introduced in the Fixed Point theorem is bounded and there exists a fixed point u ∈ X forΦ. We can then conclude that such u is a stationary solution for Problem 1. �Remark 1 Considering the Sobolev constant c0 introduced in (8), it is easy to prove that if the conditionc0 ‖ϕ‖X ′ < ν2 is satisfied, then the solution of the stationary problem is unique. This condition resembles thatof a low Reynolds number: indeed it entails also the exponential stability of the unique stationary solution ofthe evolutionary problem treated below.


4.3 The evolutionary problem

We now come to analyze the evolutionary problem. Take u ∈ L2([0, T ]; X), whose norm is

‖u‖2L2([0,T ];X)

:=T∫

0

‖u(s)‖2X ds,

and take ϕ ∈ L2([0, T ]; X ′) defined in analogy with the the previous section.

Theorem 4 For every u0 ∈ H, there exists u ∈ L2([0, T ]; X) such that

u(0) = u0, (12)∫

Ω

∂u∂t

· v + a(u, v) + b(u, u, v) = 〈ϕ, v〉, (13)

for every v ∈ L2([0, T ]; X).

Remark 2 Notice that the time derivative of u has to be interpreted as a linear form in L2([0, T ]; X ′), whoserepresentation enters Eq. (13), and thus we will take u ∈ H1([0, T ]; X ′). A key role in the evolutionary problemis played by the initial datum u0 which belongs to H . At first, the initial condition (12) should be understoodin X ′, but we will see that it actually holds in H , as u belongs to C0([0, T ]; H). After these considerations,we can take u ∈ X.

Proof In order to proceed, we need some estimates; first, we set v = u in (13) and apply Lemma 2 obtainingthat

∫

Ω

∂u∂t

· u + a(u, u) = 〈ϕ, u〉.

Integrating in time we get:

1

2

t∫

0

d

ds‖u(s)‖2

L2 ds +t∫

0

a(u, u) ds =t∫

0

〈ϕ, u〉 ds;

hence, by applying also Young’s inequality,

1

2‖u(t)‖2

L2 + ν

t∫

0

‖u‖2X ds ≤ 1

2‖u0‖2

L2 + ν

2

t∫

0

‖u‖2X ds + c2

t∫

0

‖ϕ‖2X ′ ds,

that gives the first estimate for a.e. t ∈ [0, T ]:

‖u(t)‖2L2 + ν

t∫

0

‖u‖2X ds ≤ ‖u0‖2

L2 + 2c2

t∫

0

‖ϕ‖2X ′ ds =: M.

This a priori bound tells us that any solution u of our problem belongs to a bounded subset of L2([0, T ]; X) ∩L∞([0, T ]; H). We now introduce the following theorem whose proof is given in [9, Chap. 3, Sec. 4.4].

Theorem (J. L. Lions) Let X and H be two Hilbert spaces, with X dense and continuously embedded inH; identify H with its dual in such a way that X ⊂ H ⊂ X ′ and fix T > 0. Consider a bilinear format (u, v) : X × X → R such that

(i) the function t �→ at (u, v) is measurable for every u, v ∈ X;(ii) |at (u, v)| ≤ C1 ‖u‖X ‖v‖X for a.e. t ∈ [0, T ], for every u, v ∈ X;

(iii) at (v, v) ≥ α ‖v‖2X − C2 ‖v‖2

H for a.e. t ∈ [0, T ], for every v ∈ X;

where α > 0, C1 and C2 are constants.


Then, for every f ∈ L2([0, T ]; X ′) and for every u0 ∈ H, there exists only one u such that

u ∈ L2([0, T ]; X) ∩ C0([0, T ]; H) ∩ H1([0, T ]; X ′)u(0) = u0

⟨∂u∂t

(t), v⟩

+ at (u(t), v) = 〈f(t), v〉 for a.e. t ∈ [0, T ], for every v ∈ X .

It is easy to see that the spaces X, H and the bilinear form at (u, v) := a(u(t), v(t)) fulfill the hypotheses ofthe previous theorem; then, the function

L :{

X → H × L2([0, T ]; X ′)u �→ (

u(0), ∂u∂t (t) + a(u(t), ·))

is a homeomorphism.We can now write Eqs. (12), (13) in H × L2([0, T ]; X ′) as

L(u) + (0, K0(u)) = (u0, ϕ),

from which

u = L−1(u0, ϕ − K0(u)) =: Φ(u).

By Theorem 2 and composition arguments, Φ turns out to be a compact operator, and we can then apply theFixed Point theorem in the Banach space X. Arguing as in the previous section, we take λ > 1 and assumeΦ(u) = λu; in particular u0 = λu(0) and

λ

2

t∫

0

d

ds‖u(s)‖2

L2 ds + λ

t∫

0

a(u, u) ds =t∫

0

〈ϕ, u〉 ds.

We then obtain

λ

2‖u(t)‖2

L2 + λν

t∫

0

‖u‖2X ds ≤ λ

2

∥∥λ−1u0

∥∥2

L2 + ν

2

t∫

0

‖u‖2X ds + c2

t∫

0

‖ϕ‖2X ′ ds,

that gives

λ ‖u(t)‖2L2 + (2λ − 1)ν

t∫

0

‖u‖2X ds ≤ λ−1 ‖u0‖2

L2 + 2c2

t∫

0

‖ϕ‖2X ′ ds < M.

Since (2λ − 1) > λ, we can write

‖u(t)‖2L2 + ν

t∫

0

‖u‖2X ds < λ−1 M

and the set S introduced in the Fixed Point theorem is bounded in L2([0, T ]; X) ∩ L∞([0, T ]; H). In order tocomplete the proof, it remains to show that S is bounded also in H1([0, T ]; X ′).

If there exists λ > 1 such that Φ(u) = λu, then we have that

∂u∂t

= −a(u, ·) − 1

λK0(u) + 1

λϕ in L2([0, T ]; X ′)

and that∥∥∥∥∂u∂t

∥∥∥∥

X ′≤ ‖a(u, ·)‖X ′ + 1

λ‖K0(u)‖X ′ + 1

λ‖ϕ‖X ′ .


By the continuity of a and the embeddings mentioned in the proof of Theorem 2 we can write:∥∥∥∥∂u∂t

∥∥∥∥

X ′≤ c3 ‖u‖X + c4

λ‖K0(u)‖

L32

+ 1

λ‖ϕ‖X ′ .

We know that u belongs to a bounded subset of L2([0, T ]; X) and ϕ ∈ L2([0, T ]; X ′); we deduce that

T∫

0

∥∥∥∥∂u∂t

∥∥∥∥

2

X ′≤ c5 + 2c2

4

λ2

T∫

0

‖K0(u)‖2

L32

< N

for a fixed N > 0, since K0 maps bounded subsets of L2([0, T ]; X) ∩ L∞([0, T ]; H) to bounded subsets of

L2([0, T ]; L32 (Ω; R

3)).The last bound shows that ∂u

∂t belongs to a bounded subset of L2([0, T ]; X ′) and implies that S is boundedin X. Hence there exists a fixed point u ∈ X for Φ and this is a solution for the Cauchy problem (12), (13).

�Thanks to the H2-regularity of the solution we have found, which in particular guarantees the L∞-regularity

in three dimensions, we can prove an important uniqueness result.

Theorem 5 There exists a unique solution u ∈ X of the Cauchy problem (12), (13).

Proof Let u1 and u2 be solutions of Eq. (13) with the same initial datum and set w := u1 − u2. We easilyobtain, by Eqs. (12), (13) and Lemma 2,

w(0) = 0,∫

Ω

∂w∂t

· w + a(w, w) + b(w, u2, w) = 0,

from which, integrating in time and applying the coercivity of a, we can deduce that

1

2‖w(t)‖2

L2 + ν

t∫

0

‖w‖2X ds ≤

t∫

0

|b(w, u2, w)| ds.

By Young’s inequality,

‖w(t)‖2L2 + 2ν

t∫

0

‖w‖2X ds ≤ 2

t∫

0

‖w‖L∞ ‖∇u2‖L2 ‖w‖L2 ds

≤ 2c0

t∫

0

‖w‖X ‖u2‖X ‖w‖L2 ds ≤ 2ν

t∫

0

‖w‖2X ds + c2

0

2ν

t∫

0

‖u2‖2X ‖w‖2

L2 ds;

hence we have

‖w(t)‖2L2 ≤ c2

0

2ν

t∫

0

‖u2(s)‖2X ‖w(s)‖2

L2 ds

and by Gronwall’s lemma we conclude that ‖w(t)‖2L2 = 0 for every t ∈ [0, T ], that is, u1 = u2. �

Remark 3 The previous argument can be easily adapted in order to prove the continuous dependence of thesolutions of the evolutionary problem upon initial data. In fact, if we take u1 and u2 solutions with differentinitial data, then w(0) �= 0 and we obtain


‖w(t)‖2L2 ≤ ‖w(0)‖2

L2 + c20

2ν

t∫

0

‖u2(s)‖2X ‖w(s)‖2

L2 ds;

again, by Gronwall’s lemma we conclude:

‖w(t)‖2L2 ≤ ‖w(0)‖2

L2 exp

⎛

⎝c2

0

2ν

t∫

0

‖u2(s)‖2X ds

⎞

⎠ .

Acknowledgments The authors wish to thank Marco Degiovanni and Paolo Podio-Guidugli for offering many helpful sugges-tions.

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