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Shape optimization for non-Newtonian fluids in
time-dependent domains
Jan Sokolowski, Jan Stebel
To cite this version:
Jan Sokolowski, Jan Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains.
Evolution Equations and Control Theory, 2014, 3 (2), pp.331-348. �hal-00870119�
Shape optimization for non-Newtonian fluids
in time-dependent domains
Jan Soko lowski
Institut ´
Elie Cartan, UMR 7502 Nancy-Universit´
e-CNRS-INRIA,
Laboratoire de Math´
ematiques,
Universit´
e Lorraine, B.P. 239,
54506 Vandoeuvre L`
es Nancy Cedex, France,
Systems Research Institute of the Polish Academy of Sciences,
ul. Newelska 6, 01-447 Warszawa, Poland,
Jan.Sokolowski@univ-lorraine.fr
Jan Stebel
Institute of Mathematics of the Academy of Sciences
of the Czech Republic,
ˇ
Zitn´
a 25, 110 00 Praha 1, Czech Republic,
stebel@math.cas.cz
October 5, 2013
Abstract
We study the model of an incompressible non-Newtonian fluid in a moving domain. The domain is defined as a tube built by the velocity field V and described by the family of domains Ωt parametrized by t ∈ [0, T ].
A new shape optimization problem associated with the model is defined for a family of initial domains Ω0 and admissible velocity vector fields.
It is shown that such shape optimization problems are well posed under the classical conditions on compactness of the admissible shapes [18]. For the state problem, we prove the existence of weak solutions and their continuity with respect to perturbations of the time-dependent boundary, provided that the power-law index r ≥ 11/5.
Keywords: shape optimization, time-dependent domain, incompressible viscous fluid, Navier-Stokes equations
1
Introduction
Mathematical models of fluid dynamics involving time-dependent geometry are important in many real-life applications. Devices such as automobile engines, marine and aircraft propellers, industrial mixers, peristaltic pumps, artificial hearts, to name a few, are all examples where a solid object is evolving in a predictable way and governs the motion of the fluid. Optimization of the related processes can often be achieved via improvements in the geometry or in the dynamics of the moving boundary. In many of such real systems there are present fluids, e.g. polymers, oils, lubricants, or blood, which exhibit effects that cannot be captured by the traditional Newtonian constitutive model.
In this paper we present new results on the well-posedness and domain dependence of solutions to a model of incompressible non-Newtonian fluid of power-law type confined to a time-dependent 3-dimensional domain. Subse-quently we prove the existence of optimal shapes for a broad class of shape optimization problems involving the full evolution of the geometry.
Given a bounded initial domain Ω0⊂ R3and a vector field V : [0, T ] × R3→
R3, we introduce the time-varying domain Ωt:
Ωt:= {X(t, x0); x0∈ Ω0},
∂tX(t, x0) = V(t, X(t, x0)), X(0, x0) = x0.
It is also convenient to define the space-time domain Qf := Qf(Ω0, V) and its
lateral boundary Σ := Σ(Ω0, V):
Qf := {(t, x); t ∈ (0, T ), x ∈ Ωt},
Σ := {(t, x); t ∈ (0, T ), x ∈ ∂Ωt}.
We consider the flow of an incompressible fluid governed by the following system of equations: ∂tv + div (v ⊗ v) − div S(Dv) + ∇p = f in Qf, div v = 0 in Qf, v = V on Σ, v(0, ·) = v0 in Ω0. (P (Ω0, V))
Here S stands for the traceless part of the Cauchy stress, which depends on the symmetric part Dv of the velocity gradient:
S(Dv) = ν(|Dv|2)Dv, (1) the symbol ν denotes the (generalized) viscosity and |Dv|2 is called the shear
rate. The precise constraints put on the form of S will be specified in Section 2. We shall investigate the dependence of the solutions to (P (Ω0, V)) on the
initial geometry Ω0 as well as on its time evolution described by V. For this
cO, cV > 0, and introduce the system of admissible initial domains and the
system of admissible fields:
O := {Ω0; Ω0⊂ D, Ω0∈ C1with the C1-norm being bounded by cO},
V := {V ∈ C2([0, T ]; C2
0,div(D)); kVkC2([0,T ]×D)≤ cV}.
Our motivation is to investigate two types of shape optimization problems. Optimization of initial domain. In the first case we consider flow problems where the time evolution is given and the control variable is the shape of the initial domain. This applies e.g. to optimization of shape of a rotating device when the angular velocity is a priori known. Let V ∈ V be given. We define the graph of control-to-state mapping
GI := {(Ω0, v); Ω0∈ O, v is a solution to (P (Ω0, V))}.
Consider a shape functional JI : GI → R. Then we formulate the following
problem.
Find (Ω∗0, v∗) ∈ GI such that
JI(Ω∗0, v∗) ≤ J (Ω0, v) ∀(Ω0, v) ∈ GI.
)
(PI)
Optimization of time evolution. The second case corresponds to improving the behaviour of a given initial geometry with respect to time. We fix Ω0∈ O
and define
GE:= {(V, v); V ∈ V, v is a solution to (P (Ω0, V))}.
Given a shape functional JE: GE → R, we formulate the problem:
Find (V∗, v∗) ∈ GE such that
JI(V∗, v∗) ≤ JI(V, v) ∀(V, v) ∈ GE.
)
(PE)
Here we have in mind e.g. optimization of the operating speed of a rotating or translating device.
Simultaneous optimization of shape and evolution. Problems (PI) and
(PE) can be seen as special cases of a more general task, namely to optimize
the initial shape alongside with its evolution. Let
G := {(Ω0, V, v); Ω0∈ O, V ∈ V, v is a solution to (P (Ω0, V))}
and J : G → R be a shape functional. The shape optimization problem reads: Find (Ω∗0, V∗, v∗) ∈ G such that
J (Ω∗0, V∗, v∗) ≤ J (Ω0, V, v) ∀(Ω0, V, v) ∈ G.
)
(P)
Fluids whose viscosity depends on the shear rate form an important class of non-Newtonian models (see e.g. [20, 19, 25] for general references). The first mathematical results were established more than 40 years ago in [11, 13] and [14], though the theory has been still improving (see [9, 10, 15, 5] for some of recent results). As far as the time-dependent domain is considered, the linear Newtonian model was studied already in [12] and later e.g. in [17]; we also mention results for compressible Navier-Stokes system [7, 8].
In context of optimal and boundary control, fluids with shear-dependent viscosity in a fixed domain were studied e.g. in [21, 26, 1, 22, 23]. For con-trol problems with the incompressible Navier-Stokes equations we refer to [2]. Shape optimization for Navier-Stokes equations in time-dependent domains, in particular shape sensitivity analysis, was done by [6], see also [16].
To the best of our knowledge, there are no results on the existence or domain dependence of weak solutions for the problem (P (Ω0, V)). In the analysis, two
main sources of difficulties arise: First, the space for weak solutions depends on the moving domain, thus some tools such as density of smooth functions have to be developed and the time derivative has to be introduced causiously. Secondly, the limit of the nonlinear convective term and the viscous term can be achieved provided we show strong convergence of solutions. We shall consider the so called shear-thickening case, when the weak solutions satisfy an energy equality, and the identification of the weak limit can be done using the theory of monotone operators.
The paper is organized as follows. In Section 2 we show for given (Ω0, V) ∈
O × V the existence of a weak solution to (P (Ω0, V)) and derive a uniform
es-timate independent of (Ω0, V). In this part we also mention some properties of
time-dependent domains and develop necessary tools for the functions defined in these domains. Section 3 deals with the dependence of weak solutions to (P (Ω0, V)) on the pair (Ω0, V). We show the stability of the solutions with
respect to domain perturbations of certain type. Having the result at our dis-posal, in Section 4 we easily prove the existence of minimizers for problems (PI),
(PE) and (P).
Throughout the paper, Lp(B), W1,p(B) stands for the Lebesgue and the Sobolev space, respectively, their norms are denoted by k · kp,B and k · k1,p,B.
Bold symbols denote the vector-valued objects. C∞0,div(B) is the set of smooth, compactly supported and solenoidal functions in B, Lp0,div(B) is the closure of C∞
0,div(B) in the norm k · kp,B, W 1,p
0,div(B) is the subspace of W
1,p(B) of
functions which are a.e. solenoidal and whose trace is zero.
2
Well-posedness of the state problem
In this section we will show that for a given pair (Ω0, V) ∈ O × V the problem
(P (Ω0, V)) admits a weak solution. We shall require that the nonlinear function
S satisfies
S ∈ C1(R3×3sym, R 3×3
and that there exist constants r > 1, κ ∈ {0, 1}, c1> 0, c2> 0 such that for all
A, B ∈ R3×3sym:
c1(κ + |A|2)
r−2
2 |B|2≤ S0(A) :: B ⊗ B, |S0(A)| ≤ c2(κ + |A|2) r−2
2 . (2b)
Let us mention some important consequences of this assumption. Namely, if r ≥ 2 then S is monotone in the following sense:
(S(A) − S(B)) : (A − B) > c3|A − B|r ∀A, B ∈ R3×3sym, (3)
and moreover, there is a constant c4> 0 such that
|S(A)| ≤ c4(κ + |A|)r−1 ∀A ∈ R3×3sym. (4)
As far as the initial condition is considered, we shall assume that v0∈ L2(Ω0), (v0− V(0, ·))|Ω0 ∈ L
2
0,div(Ω0). (5)
Further, the right hand side has to satisfy:
f ∈ Lr0(Qf). (6)
Now we turn to the definition of a weak solution. By a weak solution of (P (Ω0, V)) we mean any function v : Qf → R3 satisfying:
• Integrability.
the mapping t 7→ kv(t, ·)k2,Ωt belongs to L
∞(0, T ),
the mapping t 7→ kv(t, ·)k1,r,Ωt belongs to L
r(0, T ). (7a)
• Boundary condition.
(v − V)(t, ·) = 0 for a.a. t ∈ (0, T ). (7b) • Divergence free property.
div v(t, ·) = 0 for a.a. t ∈ (0, T ). (7c) • Momentum equation. For every φ ∈ C∞(Qf; R3) s.t. φ|Σ = 0,
φ(T, ·) = 0, div φ = 0, it holds: ˆ T 0 [−(v, ∂tφ)Ωt+ (S(Dv), Dφ)Ωt− (v ⊗ v, ∇φ)Ωt] = (v0, φ(0, ·))Ω0+ ˆ T 0 (f , φ)Ωt. (7d)
In what follows we shall assume that every solution v to (7) is extended by V to Qs:= ((0, T ) × D) \ Qf. Notice that due to the boundary condition (7b) we
have v ∈ Lr(0, T ; W1,r
0,div(D)). The main result of this section is stated in the
Theorem 1. Let (2) be satisfied and
r ≥11
5 . (8)
Then there is a function E : R+×R+→ R+such that for every (Ω
0, V) ∈ O ×V
and the data (v0, f ) satisfying (5)–(6), problem (P (Ω0, V)) has a weak solution
v in the sense of (7), which satisfies the estimate
sup t∈(0,T ) kv(t, ·)k2 2,Ωt+ ˆ T 0 kvkr 1,r,Ωt ≤ E(kv0k2,Ω0, kf kr0,Qf). (9)
Moreover, every solution of (7) satisfies (9).
Remark. We will show that every weak solution of (P (Ω0, V)) satisfied the
so-called energy equality, see (2.2).
Remark. For r ≥52 the weak solution is unique, see e.g. [11].
In the following two subsections we show some auxiliary results, required in the sequel. Sections 2.3–2.6 are devoted to the proof of Theorem 1. We use the penalization method: First, for a fixed ε > 0 we study a penalized system (Pε(Ω0, V)) in the fixed domain (0, T ) × D. We show uniform bounds of the
solutions, independent of ε > 0. Then we pass to the limit in the momentum equation, where all terms except for the nonlinearity S(Dv) are identified. Fi-nally, we show pointwise convergence of S(Dvε) using the monotone operator
theory.
2.1
Properties of time-dependent domains and function
spaces
The domain convergence for Dirichlet boundary conditions in fluid dynamics is considered in the monograph [18]. There, the existence of optimal shapes is shown for the compressible Navier-Stokes equations in bounded domains. We use the same framework for the shape continuity in the case of incompressible models.
Definition 1. We say that (Ωn
0, Vn) → (Ω0, V) if Ωn0 → Ω0in Hausdorff
com-plementary topology as well as with respect to the Kuratowski-Mosco convergence of the Sobolev spaces [18] and Vn→ V in C1([0, T ] × D).
The definition of O and V gives rise to the following properties of the time-dependent domains (see [24, 4, 3]):
• System O × V is compact with respect to the convergence introduced in Definition 1.
• There is a constant c > 0 such that for every (Ω0, V) ∈ O × V and
(t, x) ∈ Qf:
dist∂Ωt(x) ≤ c distΣ(x).
• For every (Ω0, V) ∈ O × V and for a.a. (t, x) ∈ Qf there exists a time
interval [T1, T2] ⊂ (0, T ) and an open set K ⊂ D such that
(t, x) ∈ [T1, T2] × K ⊂ Qf.
• Given a sequence {(Ωn
0, Vn)}∞n=1⊂ O × V such that (Ωn0, Vn) → (Ω0, V)
in O × V, then for every cylinder [T1, T2] × K ⊂ Qf there exists n0∈ N
such that
∀n ≥ n0: [T1, T2] × K ⊂ Qnf := Qf(Ωn0, V n).
For s ∈ [1, ∞) we define the space
Xs= Xs(Ω0, V) := ( φ : Qf → R3; ˆ Qf (|φ|s+ |∇φ|s) < ∞, div φ = 0, φ|Σ= 0 )
endowed with the norm
kφkXs:= ˆ Qf (|φ|s+ |∇φ|s) !1/s .
It can be easily verified that Xs is a separable Banach space, for s > 1 also reflexive. If φ ∈ Ls(Qf) is such that kφkXs < ∞, then φ(t, ·) ∈ W1,s(Ωt) for
a.a. t ∈ (0, T ), hence the trace of φ on Σ is well-defined. We shall need a density result for the space Xs. For this purpose we introduce the set
X := n X i=1 ψi(t)ϕi(x); ψi∈ D(T1i, T i 2), ϕi∈ C∞0,div(K i), [T1i, T2i] × Ki⊂ Q f, i = 1, . . . , n . (10) Lemma 1. Let s > 1. Then X is dense in Xs.
Proof. Let φ ∈ Xs and ε > 0 be arbitrary. For any λ > 0 we define the
L∞-truncation φλ(t, x) := φ(t, x) min 1, λ kφ(t, ·)kW1,s(Ω t) . Clearly, φλ ∈ L∞(0, T ; W1,s(D)), φ λ(t, ·) ∈ W 1,s
0,div(Ωt) for a.a. t ∈ (0, T ) and
moreover we can fix λ > 0 sufficiently large so that kφλ− φkXs <
ε 4.
Since for every t ∈ [0, T ], C∞0,div(Ωt) is dense in W1,s0,div(Ωt), we can find for any
• for a.a. t ∈ [0, T ]: φλ,µ(t, ·) ∈ C∞0,div(Ωt),
• for a.a. t ∈ [0, T ]: dist∂Ωtsupp φλ,µ(t, ·) > µ,
• for a.a. t ∈ [0, T ]: φλ,µ(t, ·) → φλ(t, ·) in W 1,s 0,div(Ωt).
From the Lebesgue dominated convergence theorem it follows that φλ,µ→ φλ
in Xs, µ → 0. We fix µ > 0 such that
kφλ,µ− φλkXs <
ε 4.
Due to the boundedness of ∂tV in time, it holds that distΣsupp φλ,µ > 0.
Hence, for ν > 0 sufficiently small, we can mollify φλ,µin time so that the new function φλ,µ,ν ∈ C0∞(0, T ; C∞0,div(D)), supp φλ,µ,ν ⊂ Qf and
kφλ,µ,ν− φλ,µkXs<
ε 4.
Further, for sufficiently large n ∈ N there is a covering {(T1i, T2i)}ni=1of (0, T ) and
open sets {Ki}n
i=1 such that T2i− T1i < 2/n, [T1i, T2i] × Ki ⊂ Qf, i = 1, . . . , n
and supp φλ,µ,ν ⊂ Sn
i=1[T i
1, T2i] × Ki. We take a partition of unity {ψin}ni=1
subordinate to the covering {(Ti
1, T2i)}ni=1 and define φ n i(t, x) := ψni(t) ˜φλ,µ,ν, where ˜φi,nλ,µ,ν(x) :=fflT2i Ti 1 φλ,µ,ν(τ, x) dτ . ThenPn i=1φ n i ∈ X . It remains to show thatPn i=1φ n
i → φλ,µ,ν as n → ∞. The mean value theorem implies that
kψniφλ,µ,ν− φ n ik s Xs≤ ˆ D ˆ Ti 2 Ti 1 |φλ,µ,ν− ˜φ i,n λ,µ,ν| s+ |∇φ λ,µ,ν− ∇ ˜φ i,n λ,µ,ν| s ≤ |T2i− T i 1| s ˆ D ˆ T2i Ti 1 |∂tφλ,µ,ν| s+ |∂ t∇φλ,µ,ν| s ≤ |Ti 2− T1i|s+1 k∂tφλ,µ,νks∞,D+ k∂t∇φλ,µ,νks∞,D ≤ C ns+1. Hence, kφλ,µ,ν− n X i=1 φnikXs ≤ n X i=1 kψinφλ,µ,ν − n X i=1 φnikXs ≤ Cn−1/s.
Choosing n > (C/ε)s we conclude that
kφ −
n
X
i=1
It is easy to see that for any φ ∈ Xsand g ∈ (Xs)∗, hg, φiXs = ˆ T 0 hg(t), φ(t, ·)iW1,s 0,div(Ωt)dt.
Consequently, due to Lemma 1, we can introduce the notion of time derivative for functions from Xs.
Definition 2. Let s > 1 and u ∈ Ls0(Q
f). We say that g ∈ (Xs)∗is the (weak)
time derivative of u if ˆ T 0 ψhg, ϕiW1,s 0,div(Ωt)= − ˆ T 0 ψ0(u, ϕ)D for all (ψ, ϕ) ∈ D(0, T ) × C∞
0,div(D) such that supp(ψϕ) ⊂ Qf. As usual, we
denote ∂tu := g.
The time derivative of a function defined in the time-dependent domain shares many properties of the usual time derivative in Bochner spaces. We mention some of them.
• For any s ≥ 6 5 and u ∈ X s s.t. ∂ tu ∈ (Xs)∗ we have: h∂tu, uiW1,s0,div(Ωt)= 1 2 d dtkuk 2 2,Ωt in D 0(0, T ). (11)
• The momentum equation (7d) holds due to the density for any test func-tion φ ∈ Xrin the following form:
h∂tv, φiXr+ ˆ T 0 [(S(Dv), Dφ)Ωt− (v ⊗ v, ∇φ)Ωt] = ˆ T 0 (f , φ)Ωt. (12)
2.2
Energy equality
As a consequence of the condition (8), one can use φ := χ[0,τ ](v−V), τ ∈ (0, T ),
as a test function in (12). The first term can be rearranged as follows:
h∂tv, χ[0,τ ](v − V)iXr = ˆ τ 0 h∂t(v − V), v − ViW1,r 0,div(Ωt) + ˆ τ 0 (∂tV, v)Ωt− ˆ τ 0 (∂tV, V)Ωt. (13)
From the Reynolds transport theorem and the fact that V is the velocity of the time-dependent boundary ∂Ωt, we get:
1 2 d dtkVk 2 2,Ωt = (∂tV, V)Ωt+ 1 2(V · n, |V| 2 )∂Ωt = (∂tV, V)Ωt− 1 2(V · n, |V| 2) D\∂Ωt = (∂tV, V)Ωt− (v ⊗ v, ∇V)D\Ωt. (14)
This identity and (11) enables to replace the last term in (13), which yields: h∂tv, χ[0,τ ](v − V)iXr = 1 2kv(τ, ·)k 2 2,Ωτ − 1 2kv0k 2 2,Ω0 + (v0, V(0, ·))Ω0− (v(τ, ·), V(τ, ·))Ωτ + ˆ τ 0 (v, ∂tV)Ωt − ˆ τ 0 (v ⊗ v, ∇V)D\Ωt. (15)
Next, using the fact that v = V in Qsand the well known identity
∀u ∈ W1,6/50,div(D) : (u ⊗ u, ∇u)D= 0, (16)
we can rewrite the convective term arising from (12) in the form
− ˆ τ 0 (v ⊗ v, ∇(v − V))D= ˆ τ 0 (v ⊗ v, ∇V)Ωt+ ˆ τ 0 (v ⊗ v, ∇V)D\Ωt. (17)
Relations (12), (15) and (17) then lead to the following energy equality, which holds for a.a. τ ∈ (0, T ):
1 2kv(τ )k 2 2,Ωτ + ˆ τ 0 (S(Dv), Dv)Ωt = 1 2kv0k 2 2,Ω0+ (v(τ ), V(τ, ·))Ωτ − (v0, V(0, ·))Ω0+ ˆ τ 0 (S(Dv), DV) Ωt− ˆ τ 0 (v ⊗ v, ∇V)Ωt + ˆ τ 0 (f , v − V)Ωt− ˆ τ 0 (v, ∂tV)Ωt. (18)
2.3
Penalized problem
We extend f by zero to Qs, so that the resulting function, denoted by the same
symbol f ∈ Lr0((0, T ) × D). Next, we extend v0 by V(0, ·) to D \ Ω0, which
implies that v0 ∈ L20,div(D). For a given ε > 0 we consider the penalized
problem of the form
∂tv + div (v ⊗ v) − div S(Dv) + ∇p + 1 εχ(v − V) = f in (0, T ) × D, div v = 0 in (0, T ) × D, v = 0 on (0, T ) × ∂D, v(0, ·) = v0 in D. (Pε(Ω0, V))
Here χ denotes the characteristic function of the set Qs, i.e.
χ(t, x) = (
1 if (t, x) ∈ Qs,
Since problem (Pε(Ω0, V)) is posed in the fixed domain and the term 1εχ(v − V)
is a compact perturbation, we can use the available results, e.g. [5, Theorem 1.3], to obtain the existence of a weak solution to (Pε(Ω0, V)), denoted vε,
which satisfies: • vε∈ Lr(0, T ; W
1,r
0,div(D)) ∩ L
2(0, T ; L2(D));
• for every φ ∈ C∞([0, T ); C∞0,div(D)): ˆ T 0 − (vε, ∂tφ)D+ (S(Dvε), Dφ)D− (vε⊗ vε, ∇φ)D +1 ε(χ(vε− V), φ)D = (v0, φ(0, ·))D+ ˆ T 0 (f , φ)D. (19)
Note that due to (8), every solution satisfies for a.a. τ ∈ (0, T ) the energy equality: 1 2kvε(τ )k 2 2,D+ ˆ τ 0 (S(Dvε), Dvε)D+ 1 ε(χ(vε− V), vε)D =1 2kv0k 2 2,D+ ˆ τ 0 (f , vε)D. (20)
2.4
Modified energy equality and uniform estimates
Due to its properties, we can use V as a test function in (19) and subtract the result from (20), so that we obtain for a.a. τ ∈ (0, T ) the following modified energy equality: 1 2kvε(τ )k 2 2,D+ ˆ τ 0 (S(Dvε), Dvε)D+ 1 ε ˆ D χ|vε− V|2 =1 2kv0k 2 2,D+ (vε(τ ), V(τ, ·))D− (v0, V(0, ·))D+ ˆ τ 0 (S(Dvε), DV)D − ˆ τ 0 (vε⊗ vε, ∇V)D+ ˆ τ 0 (f , vε− V)D− ˆ τ 0 (vε, ∂tV)D. (21)
In order to obtain bounds for vε that are independent of ε > 0, Ω0 ∈ O and
V ∈ V, we estimate the right hand side of (21) from above, using eventually (4), H¨older’s and Young’s inequality. In particular we have:
(vε(τ ), V(τ, ·)) ≤ αkvεk22,D+ c(α)kV(τ, ·)k 2 2,D, |(v0, V(0, ·))| ≤ 1 2kv0k 2 2,D+ 1 2kV(0, ·)k 2 2,D, ˆ τ 0 (S(Dvε), DV)D≤ β ˆ τ 0 kvεkr1,r,D+ c(β, c4, r) ˆ T 0 kVkr 1,r,D+ c(β, c4, κ, r),
ˆ τ 0 (vε⊗ vε, ∇V)D ≤ ˆ τ 0 k∇Vk∞,Dkvεk22,D, ˆ τ 0 (f , vε− V)D≤ β ˆ τ 0 kvεkr1,r,D+ c(β, r)kf kr 0 r0,Q f + ˆ T 0 kVkr 1,r,D, ˆ τ 0 (vε, ∂tV)D ≤ 1 2 ˆ τ 0 kvεk22,D+ 1 2 ˆ T 0 k∂tVk22,D,
where α, β > 0 are arbitrary numbers to be specified later. To the second term on the left hand side of (21) we apply (3) and Korn’s inequality on W1,r0 (D):
ˆ τ 0 (S(Dvε), Dvε)D≥ CKc3 ˆ τ 0 kvεkr1,r,D.
Here CK> 0 is the constant of the Korn inequality. Using the above estimates
we obtain from (21): 1 2− α kvε(τ )k22,D+ (CKc3− 2β) ˆ τ 0 kvεkr1,r,D+ 1 εkvε− Vk 2 2,Qs ≤ ˆ τ 0 (1 + k∇Vk∞,D)kvεk22,D+ kv0k22,D+ c(β, r)kf k r0 r0,Q f + (1 + c(β, c4, r)) ˆ τ 0 kVkr1,r,D+ 1 2k∂tVk 2 2,(0,T )×D +1 2kV(0, ·)k 2 2,D+ c(α)kV(τ, ·)k 2 2,D+ c(β, c4, κ, r).
We set α := 14, β := 14CKc3and estimate the terms depending solely on v0, f and
V by a constant C := C(kv0k2,Ω0, kf kr0,Qf, kVkC1([0,T ]×D), |D|, r, κ, c3, c4) > 0, which yields: 1 4kvε(τ )k 2 2,D+ 1 2CKc3 ˆ τ 0 kvεkr1,r,D+ 1 εkvε− Vk 2 2,Qs ≤ ˆ τ 0 (1 + k∇Vk∞,D)kvεk22,D+ C. (22)
Applying Gronwall’s inequality we conclude that
sup τ ∈(0,T ) kvε(τ, ·)k22,D+ ˆ T 0 kvεkr1,r,D+ 1 εkvε− Vk 2 2,Qs ≤ E, (23) where E := E(kv0k2,Ω0, kf kr0,Qf, kVkC1([0,T ]×D), |D|, T, r, κ, c3, c4, CK) > 0.
Using this information and (19) one can estimate the time derivative:
{∂tvε}ε>0 is bounded in (Xr)∗. (24)
By interpolation one also gets from (23) that
2.5
Limit ε → 0
From the uniform estimate (23) it directly follows that there exists ˆv ∈ L∞(0, T ; L2(D))∩
Lr(0, T ; W1,r 0,div(D)) and S ∈ L r0 ((0, T ) × D; R3×3) such that vε*∗vˆ in L∞(0, T ; L2(D)), (26) vε* ˆv in Lr(0, T ; W 1,r 0,div(D)), (27) S(Dvε) * S in Lr 0 ((0, T ) × D; R3×3), (28) vε→ V in L2(Qs), (29)
passing to a subsequence, as the case may be. Consequently ˆv satisfies the boundary condition ˆv = V on Σ. Due to (25) we have:
vε⊗ vε* v ⊗ v in L5r/6((0, T ) × D).
Here and in what follows, H(v) denotes a weak limit of a compound function H(vε) or H(vn). Estimates (23), (24) together with the Aubin-Lions lemma
imply that
vε→ ˆv strongly in Ls((T1, T2) × K), s ∈ [1, 5r/3), (30)
whenever [T1, T2] × K ⊂ Qf. This together with (29) and the facts mentioned
in Section 2.1 yields:
vε→ ˆv a.a. in (0, T ) × D (31)
and consequently
v ⊗ v = ˆv ⊗ ˆv a.a. in (0, T ) × D.
We have derived convergence which enables to pass to the limit with ε → 0+ in the weak formulation. In particular, we have for all φ ∈ Xr:
h∂tv, φiˆ Xr+ ˆ T 0 (S, Dφ)Ωt− (ˆv ⊗ ˆv, ∇φ)Ωt = ˆ T 0 (f , φ)Ωt. (32)
In the next part we shall prove that S = S(Dˆv).
2.6
Convergence of the nonlinear term S(Dv
ε)
We shall use the monotonicity of S to show the pointwise convergence of Dvε.
Let ψ ∈ D(0, T ), ψ ≥ 0. From (3) we have:
c3 ˆ T 0 ψkD(vε− ˆv)krr,D≤ ˆ T 0 ψ(S(Dvε) − S(Dˆv), D(vε− ˆv))D = ˆ T 0 ψ(S(Dv ε), D(vε− V))D+ ˆ T 0 ψ(S(Dv ε), D(V − ˆv))D − ˆ T 0 ψ(S(Dˆv), D(vε− ˆv))D.
In the last two terms one can passing to the limit: ˆ T 0 ψ(S(Dvε), D(V − ˆv))D→ ˆ T 0 ψ(S, D(V − ˆv))D, ˆ T 0 ψ(S(Dˆv), D(vε− ˆv))D→ 0, ε → 0 + .
In order to proceed in the first term we observe that due to (19) and Lemma 1, we have for any φ ∈ Xr:
ˆ T 0 h∂tvε, φiW1,r0,div(D)+ (S(Dvε), Dφ)D− (vε⊗ vε, ∇φ)D +1 ε(χ(vε− V), φ)D = ˆ T 0 (f , φ)D. (33)
Hence, by choosing φ := ψ(vε− V) we obtain:
ˆ T 0 ψ(S(Dv ε), D(vε− V))D= − ˆ T 0 ψh∂tvε, vε− ViW1,r 0,div(D) + ˆ T 0 ψ [(vε⊗ vε, ∇(vε− V))D+ (f , vε− V)D] − 1 εkvε− Vk 2 2,Qs.
and, in accordance with (16),
lim ε→0+ ˆ T 0 ψ(S(Dvε), D(vε− V))D≤ − lim ε→0+ ˆ T 0 ψh∂tvε, vε− ViW1,r 0,div(D) + ˆ T 0 ψ [(ˆv ⊗ ˆv, ∇(ˆv − V))D+ (f , ˆv − V)D] . (34)
We rewrite the term from (34) containing the time derivative:
− ˆ T 0 ψh∂tvε, vε− ViW1,r 0,div(D) = 1 2 ˆ T 0 ψ0kvεk22,D − ˆ T 0 (vε, ∂t(ψV))D.
From (31), (25), the fact that 5r3 > 2 and the Vitali lemma it follows that ˆ T 0 ψ0kvεk22,D→ ˆ T 0 ψ0kˆvk22,D, ε → 0 + . Hence, − ˆ T 0 ψh∂tvε, vε− ViW1,r 0,div(D)→ 1 2 ˆ T 0 ψ0kˆvk22,D− ˆ T 0 (ˆv, ∂t(ψV))D = − ˆ T 0 ψh∂tv, ˆˆ v − ViW1,r 0,div(Ωt), ε → 0 + .
Using all the above computations, the fact that ˆv = V in Qsand (32) we obtain: ˆ T 0 ψ(S(Dv ε), D(vε− V))D→ ˆ T 0 ψ(S, D(ˆ v − V))D, ε → 0+, and consequently ˆ T 0 ψkD(vε− ˆv)krr,D→ 0, ε → 0 + .
Therefore, Dvε→ Dˆv pointwise, the Vitali lemma yields:
S = S(Dˆv) a.a. in (0, T ) × D
and thus v := ˆv|Qf is a weak solution. The fact that every weak solution is
bounded in the sense of (9) can be verified by using (18) together with the arguments from Section 2.4. Theorem 1 is therefore proved.
3
Domain dependence of weak solutions
In this section we shall consider a sequence of problems {(P (Ωn
0, Vn))}∞n=1with
the initial conditions vn0 and the force terms fn. The sequence of functions {(vn
0, f n)}∞
n=1 is assumed to obey the following hypothesis:
vn0 ∈ L 2 0,div(D), v n 0 = V n (0, ·) in D \ Ωn0, (35) fn∈ Lr0((0, T ) × D; R3), fn= 0 in ((0, T ) × D) \ Qnf, n ∈ N. (36) Here Qnf denotes the space-time domain occupied by the fluid, determined by the pair (Ωn0, Vn). It can be verified that the restriction of v0n to Ωn0 satisfies
(5), namely (vn0− Vn(0, ·)) ∈ L20,div(Ω n
0). We prove the following result.
Theorem 2. Let (2) be satisfied with some r ≥115, {(Ωn
0, Vn)}∞n=1⊂ O × V be
such that
(Ωn0, Vn) → (Ω0, V) in the sense of Definition 1, (37)
the data {(vn
0, fn)} satisfy (35)–(36) and in addition:
vn0 * v0 in L2(D), fn * f in Lr
0
((0, T ) × D). (38) Let vn be a weak solution to (P (Ωn0, Vn)) with the initial condition vn0 and the force fn, n ∈ N. If vn is extended by Vn to ((0, T ) × D) \ Qf(Ωn0, Vn), then
{vn}∞
n=1 contains a subsequence (denoted by the same symbol) which satisfies:
vn*∗v in Lˆ ∞(0, T ; L2(D)),
vn* ˆv in Lr(0, T ; W1,r0,div(D)). (39) In addition, v := ˆv|Qf is a weak solution to (P (Ω0, V)) with the initial condition
v0 and the force f .
The structure of the proof is similar to the one of Theorem 1. Here the situation is however more involved since every problem (P (Ωn
0, Vn)) is posed in
3.1
Uniform bounds and weak convergence
The estimate (9) and the fact that vn was extended by Vn imply that
{vn}∞
n=1is bounded in L∞(0, T ; L
2(D)) ∩ Lr(0, T ; W1,r
0,div(D)). (40)
By interpolation one gets: {vn}∞
n=1is bounded in L
5r/3((0, T ) × D). (41)
From this information and (7d) it follows that k∂tvnk(Xr(Ωn
0,Vn))∗ ≤ c,
where c > 0 is independent of n ∈ N. In accordance with Section 2.1, for any T1, T2 ∈ (0, T ) and any open set K ⊂ D such that [T1, T2] × K ⊂ Qf, there is
a number n0 such that for n ≥ n0, the set [T1, T2] × K is contained also in Qnf.
Hence we obtain:
{∂tvn}∞n=n0 is bounded in (L
r(T
1, T2; W1,r0,div(K)))∗. (42)
The above bounds give rise, passing eventually to a subsequence, to the following convergence: vn *∗vˆ in L∞(0, T ; L2(D)), (43) vn * ˆv in Lr(0, T ; W1,r0,div(D)), (44) vn⊗ vn * v ⊗ v in L5r/6((0, T ) × D), (45) S(Dvn) * S in Lr 0 ((0, T ) × D; R3×3). (46) Due to (40), (42) and the Aubin-Lions lemma we have:
vn → ˆv in Ls((T1, T2) × K), s ∈ [1, 5r/3). (47)
This and (37) implies that
vn→ ˆv a.a. in (0, T ) × D, vˆ|Qs = V. (48)
Consequently,
v ⊗ v = ˆv ⊗ ˆv a.a. in (0, T ) × D.
3.2
Limit n → ∞
We choose φ ∈ X , which is an admissible test function for (P (Ω0, V)). For
sufficiently large n it also holds that supp φ ⊂ Qn
f. Therefore φ can be used as
a test function in (P (Ωn
0, Vn)) as well, and the function vn satisfies:
ˆ T 0 [−(vn, ∂tφ)D+ (S(Dvn), Dφ)D− (vn⊗ vn, ∇φ)D] = (vn0, φ(0, ·))D+ ˆ T 0 (fn, φ)D. (49)
With help of the convergence derived in the preceding section and (38), we can pass to the limit in (49) and, arguing by density, obtain for any φ ∈ Xr(Ω0, V):
h∂tv, φiˆ Xr+ ˆ T 0 (S, Dφ)D− (ˆv ⊗ ˆv, ∇φ)D = ˆ T 0 (f , φ)D. (50)
To see that ˆv|Qf is a weak solution to (P (Ω0, V)), it remains to show that
S = S(Dˆv) a.a. in Qf.
3.3
Identification of S
Let ψ ∈ D(0, T ), ψ ≥ 0. Similarly as in Section 2.6 we write:
c3 ˆ T 0 ψkD(vn− v)kr r,D≤ ˆ T 0 ψ(S(Dvn) − S(Dv), D(vn− v))D = ˆ T 0 ψ(S(Dv n ), D(vn− Vn)) D+ ˆ T 0 ψ(S(Dv n ), D(Vn− v))D − ˆ T 0 ψ(S(Dv), D(vn− v))D.
By a density argument, we realize that (49) can be generalized for any φ ∈ Xr(Ωn 0, Vn): h∂tvn, φiXr(Ωn 0,Vn)+ ˆ T 0 [(S(Dvn), Dφ)D− (vn⊗ vn, ∇φ)D] = ˆ T 0 (fn, φ)D. (51) Hence, by choosing φ := ψ(vn− Vn) we get:
ˆ T 0 ψ(S(Dvn), D(vn− Vn)) D= − ˆ T 0 ψh∂tvn, vn− VniW1,r 0,div(Ωnt) + ˆ T 0 ψ [(vn⊗ vn, ∇(vn− Vn))D+ (fn, vn− Vn)D] . (52)
Using (16), we can pass to the limit in (52):
lim n→∞ ˆ T 0 ψ(S(Dvn), D(vn− Vn)) D= − lim n→∞ ˆ T 0 ψh∂tvn, vn− VniW1,r 0,div(Ω n t) + ˆ T 0 ψ [(ˆv ⊗ ˆv, ∇(ˆv − V))D+ (f , ˆv − V)D] . (53)
We rewrite the term from (53) containing the time derivative:
− ˆ T 0 ψh∂tvn, vn−VniW1,r0,div(Ωn t)= 1 2 ˆ T 0 ψ0kvnk2 2,D− ˆ T 0 (vn, ∂t(ψVn))D.
From (48), (41), the fact that 5r3 > 2 and the Vitali lemma it follows that ˆ T 0 ψ0kvnk22,D → ˆ T 0 ψ0kˆvk22,D, n → ∞. Hence, − ˆ T 0 ψh∂tvn, vn− VniW1,r 0,div(Ωnt)→ 1 2 ˆ T 0 ψ0kˆvk22,D− ˆ T 0 (ˆv, ∂t(ψV))D = − ˆ T 0 ψh∂tv, ˆˆ v − ViW1,r0,div(Ωt), n → ∞.
Using all the above computations and (50) we obtain: ˆ T 0 ψ(S(Dvn), D(vn− Vn)) D→ ˆ T 0 ψ(S, D(ˆv − V))D, n → ∞, and consequently ˆ T 0 ψkD(vn− ˆv)krr,D→ 0, n → ∞. Therefore, Dvn → Dˆ
v a.a. in (0, T ) × D and by Vitali’s lemma, S = S(Dˆv). Finally, v := ˆv|Qf is a weak solution to (P (Ω0, V)).
4
Existence of optimal shapes and applications
In this section we apply the result of Theorem 2 to the existence of minimizers to (P), (PI) and (PE). In order to satisfy the hypotheses (35) and (36), we shall
take the initial conditions and forces of the following form: For a given pair ( ¯Ω0, ¯V) ∈ O × V we choose the initial condition ¯v0 satisfying (5) and the force
¯f := ˆf
|Qf( ¯Ω0, ¯V), where ˆf ∈ L
r0((0, T ) × D). For any (Ω
0, V) ∈ O × V there is
a diffeomorphism T : D → D such that T( ¯Ω0) = T(Ω0). We then use Piola’s
transform to define
v0(Ω0, V) := ∇T(¯v0− ¯V(0, ·)) ◦ T−1+ V(0, ·). (54)
The force f (Ω0, V) is defined by
f (Ω0, V) := ˆf|Qf(Ω0,V). (55)
We shall also require lower semicontinuity of the considered shape functionals. Definition 3. We say that J : G → R is lower semicontinuous if for ev-ery sequence {(Ωn 0, Vn, vn)}∞n=1 ⊂ G such that (Ωn0, Vn) → (Ω0, V), vn * v in Lr(0, T ; W1,r 0,div(D)) and v n*∗v in L∞(0, T ; L2(D)), it holds: J (Ω0, V, v) ≤ lim inf n→∞ J (Ω n 0, V n, vn).
Theorem 3. Let (8) hold for some r ≥ 11/5, the data be given by (54)–(55) and J be lower semicontinuous. Then problem (P) has a solution.
Proof. Let {(Ωn0, Vn, vn)}∞n=1be a minimizing sequence for J . Due to
compact-ness of O × V, we can extract a subsequence of {(Ωn0, Vn)}∞n=1(denoted by the
same symbol) such that
(Ωn0, Vn) → (Ω0, V).
Due to (54)–(55) and the properties of the Piola transform, the initial conditions and forces are compatible with the hypotheses (35), (36) and satisfy (38). From Theorem 2 it follows that
vn * v in Lr(0, T ; W1,r(D)), vn *∗v in L∞(0, T ; L2(D)). The lower semicontinuity of J then implies:
J (Ω0, V, v) ≤ lim inf n→∞ J (Ω n 0, V n, vn), therefore (Ω0, V, v) is a solution of (P).
The lower semicontinuity of JI and JE can be introduced in the same way
as in Definition 3. As a consequence, an analogy of Theorem 3 for (PI), (PE),
respectively, is obtained.
Conclusion
In the paper the new shape optimization problems are considered for incom-pressible fluids. The framework of domain dependence in fluid dynamics intro-duced in [18] is used for the purposes of well posedness of such problems. The proof is based on the existence and stability with respect to domain of the weak solutions to the state problem. These results are obtained for the case when the power-law index r ≥ 11/5. Extending the results to the case r ∈ (6/5, 11/5) would require to adopt some special techniques such as Lipschitz truncation and local pressure representation to the time-dependent domain setting. We believe that this extension is possible. The shape sensitivity analysis will be also considered in the following research.
Acknowledgement. The work of J. Stebel was supported by the Czech Sci-ence Foundation grant GA201/09/0917 and RVO: 67985840.
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