www.elsevier.com/locate/jde
Determination of viscosity in the stationary
Navier–Stokes equations
Xiaosheng Li
a, Jenn-Nan Wang
b,∗,1aDepartment of Mathematics, University of Washington 354350, Seattle, WA 98195-4350, USA bDepartment of Mathematics, Taida Institute of Mathematical Sciences and NCTS (Taipei),
National Taiwan University, Taipei 106, Taiwan
Received 24 January 2007 Available online 10 September 2007
Abstract
In this paper we consider the stationary Navier–Stokes equations in a bounded domain with a variable viscosity. We prove that one can uniquely determine the viscosity function from the knowledge of boundary data.
©2007 Elsevier Inc. All rights reserved.
1. Introduction
In this work we consider the unique determination of the viscosity in an incompressible fluid described by the stationary Navier–Stokes equations. Let Ω ⊂ R3 be an open bounded domain with boundary ∂Ω∈ C∞. Assume that Ω is filled with an incompressible fluid. Let
u= (u1, u2, u3)T be the velocity vector field satisfying the stationary Navier–Stokes system
div σμ(u, p)− (u · ∇)u = 0 in Ω,
div u= 0 in Ω, (1.1)
where
σμ(u, p)= 2μ Sym(∇u) − pI
* Corresponding author.
E-mail addresses: xli@math.washington.edu (X. Li), jnwang@math.ntu.edu.tw (J.-N. Wang).
1 The author was supported in part by the National Science Council of Taiwan (NSC 95-2115-M-002-003). 0022-0396/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
and Sym(A)= (A + AT)/2 is the symmetric part of the matrix A. Here μ(x) > 0 is the viscosity
function. The exact regularity of μ will be specified as we go along. Precisely, we can write the first equation of (1.1) componentwise, i.e.,
div σμ(u, p)− (u · ∇)u
k = j ∂j 2μ Sym(∇u)j k− ∂ kp− j uj∂juk.
Here and below, all Roman indices are from 1 to 3 unless otherwise indicated. The second equa-tion of (1.1) is the well-known incompressibility condiequa-tion.
We are interested in the inverse problem in this paper. We first define the meaning of boundary measurements. Mathematically, the boundary measurements are encoded in the Cauchy data of all solutions satisfying (1.1). Precisely, we define
˜Sμ:=
u|∂Ω, σμ(u, p)n|∂Ω
,
where (u, p) solves (1.1) with well-defined boundary traces u|∂Ωand σμ(u, p)n|∂Ω, n is the unit
outer normal of the ∂Ω, and u|∂Ωsatisfies the compatibility condition
∂Ω
u|∂Ω· n ds = 0. (1.2)
We want to remark that a solution (u, p) satisfies (1.1) with nonhomogeneous Dirichlet condition
u|∂Ω is not necessarily unique (see [23]). In the physical sense, σμ(u, p)n|∂Ω represents the
Cauchy force acting on ∂Ω. The inverse problem now is to determine μ from the knowledge of ˜Sμ.
To study our inverse problem, we will not consider the general Dirichlet data u|∂Ω= φ.
In-stead, we shall take
φ= εψ (1.3)
with|ε| sufficiently small and ψ ∈ H3/2(∂Ω)satisfying the compatibility condition (1.2). For such a choice of Dirichlet data, we can show that there exists a solution (u, p) to (1.1) with
u|∂Ω= εψ and the boundary trace σμ(u, p)n|∂Ω∈ H1/2(∂Ω). Thus the Cauchy data ˜Sμ is
meaningful in this case. When|ε| is sufficiently small, we even know that the solution (u, p) to (1.1) is unique (p is unique up to a constant), but we do not need it. The main result of this paper is the following global uniqueness theorem of the inverse problem.
Theorem 1. Assume that μ1(x) and μ2(x) are two viscosity functions satisfying μ1, μ2 ∈
Cn0( ¯Ω) for n0 8 and
∂αμ1(x)= ∂αμ2(x) ∀x ∈ ∂Ω, |α| 1. (1.4) Let ˜Sμ1 and ˜Sμ2 be the Cauchy data associated with μ1and μ2, respectively. If ˜Sμ1= ˜Sμ2 then
μ1= μ2.
When the boundary ∂Ω is convex and has nonvanishing Gauss curvature, we can remove the assumption (1.4) from Theorem 1.
Theorem 2. Let ∂Ω be convex with nonvanishing Gauss curvature. Assume that μ1(x) and μ2(x) are two viscosity functions satisfying μ1, μ2∈ Cn0( ¯Ω) for n0 8. If ˜Sμ1= ˜Sμ2 then μ1= μ2.
The parameter determination problem by boundary measurements is a rather well-studied field. Since Calderón’s pioneer contribution [1], a key method has been the construction of com-plex geometrical optics solutions with a large parameter which was introduced by Sylvester and Uhlmann [22] and has become a standard method. We also refer the readers to Uhlmann’s survey article [24]. The inverse problem for incompressible fluid governed by the Stokes equations was studied by Heck, Li and Wang [7]. They proved a global identifiability of the viscosity para-meter by boundary measurements using the method introduced by Eskin and Ralston [3–5] and also related work [16,17]. To study the Navier–Stokes equations we shall apply the linearization method. This method was first introduced by Isakov [9] in a semilinear parabolic inverse prob-lem. This technique allows for the reduction of the semilinear inverse boundary value problem to the corresponding linear one. The linearization strategy has been used by many authors to treat the inverse problem for nonlinear equations, see for example [8,10–15,18–21].
The difficulty in implementing the linearization technique to our problem lies in the existence of particular solutions to (1.1) which possess some controlled asymptotic properties. This is why we introduce the Dirichlet condition with small parameter ε as in (1.3). The key step in the proofs of Theorems 1 and 2 is to show that there exists a solution (uε, pε)to (1.1) with boundary
con-dition (1.3) and (ε−1uε, ε−1pε)converges to (v0, q0)in suitable Sobolev spaces, where (v0, q0)
satisfies the Stokes equations. Subsequently, one can determine the Cauchy data associated with the Stokes equations from ˜Sμ. Then the inverse problem for the Navier–Stokes equations (1.1) is
reduced to the same problem for the Stokes equations. We would like to mention that Theorems 1 and 2 for Navier–Stokes equations are counterparts of Theorem 1.1 and Corollary 1.4 in [7] for Stokes equations.
This paper is organized as follows. In Section 2, we will prove the existence of the boundary value problem for (1.1). In Section 3, we linearize the Cauchy data ˜Sμ and prove Theorems 1
and 2. In Appendix A we provide the existence, uniqueness and regularity results of the solution of the Stokes equations which we use in Section 2.
2. Direct problem
In this section we aim to prove the existence of the boundary value problem ⎧ ⎪ ⎨ ⎪ ⎩ divσμ(u, p) − (u · ∇)u = 0 in Ω, div u= 0 in Ω, u= φ ∈ H3/2(Ω) on ∂Ω (2.1)
with the compatibility condition (1.2). When μ is a constant, this problem has been well docu-mented in the literature, see for example [6,23]. Here we study the case where μ is a function and the boundary value contains a small parameter.
As mentioned in the introduction, we choose φ= εψ with ψ ∈ H3/2(∂Ω) and look for
(uε, pε)= (εvε, εqε)satisfies (2.1). The problem (2.1) is reduced to
⎧ ⎪ ⎨ ⎪ ⎩ divσμ(vε, qε) − ε(vε· ∇)vε= 0 in Ω, div vε= 0 in Ω, vε= ψ on ∂Ω. (2.2)
We will solve for (2.2) with the form vε= v0+ εv and qε= q0+ εq, where (v0, q0)satisfies the Stokes equations ⎧ ⎪ ⎨ ⎪ ⎩ divσμ(v0, q0) = 0 in Ω, div v0= 0 in Ω, v0= ψ on ∂Ω, (2.3) and (v, q) satisfies ⎧ ⎨ ⎩ − divσμ(v, q) + ε(v0· ∇)v + ε(v · ∇)v0+ ε2(v· ∇)v = f in Ω, div v= 0 in Ω, v= 0 on ∂Ω (2.4) with f = −(v0· ∇)v0.
For (2.3), we know (see Theorem 11 in Appendix A) that for each ψ∈ H3/2(∂Ω)there exists a unique (v0, q0)∈ H2(Ω)× H1(Ω)(q0 is unique up to a constant) satisfying (2.3) and the
estimate
v0H2(Ω)+ q0H1(Ω)/R CψH3/2(∂Ω), (2.5)
where q0H1(Ω)/R := infc∈Rq0+ cH1(Ω). In view of the Sobolev imbedding theorem
H2(Ω) → C0( ¯Ω), we have that
f H1(Ω)= (v0· ∇)v0 H1(Ω) Cv02H2(Ω) Cψ2H3/2(∂Ω). (2.6)
Now we need to solve (2.4). We first prove an existence theorem.
Theorem 3. There exists a positive number ε0 depending on ψ such that for any |ε| ε0,
(2.4) has at least one weak solution (v, q)∈ H01(Ω)× L2(Ω).
Proof. As outlined in [23] for the standard Navier–Stokes equations, we solve (2.4) by the
Galerkin method. Let us denote
V =v∈ H01(Ω): div v= 0.
Combining Korn’s inequality and Poincaré’s inequality, H01(Ω)is a separable Hilbert space with respect to the inner product
u, w =
Ω
S(u)· S( ¯w) dx. (2.7) Note that V is a closed subspace of H01(Ω), which is also separable. Let w1, w2, . . .be elements
ofV which form a complete orthonormal system of V , where
Let vn= n j=1 ξj,nwj with ξj,n∈ C (2.8) satisfy μvn, wj + εb(v0, vn, wj)+ εb(vn, v0, wj)+ ε2b(vn, vn, wj)= (f, wj) (2.9) for j= 1, . . . , n, where b(u, v, w)= Ω (u· ∇)v · ¯w dx and (f, w)= Ω f · ¯w dx.
Arguing as in [23, Lemma 1.3, Chapter II], one can easily prove two properties of b(u, v, w):
b(u, v, v)= 0 for all u ∈ V, v ∈ H01(Ω) (2.10) and
b(u, v, w)= −b(u, w, v) for all u ∈ V, v, w ∈ H01(Ω). (2.11) Moreover, using the imbedding H2(Ω) → C0( ¯Ω), we can see that
b(v, v, v0) = Ω (v· ∇)v · v0dx Cv0H2(Ω)v2 H1(Ω). (2.12)
Next we recall a technical lemma proved in [23].
Lemma 4. (See [23, Lemma 1.4, Chapter II].) Let X be a finite-dimensional Hilbert space with
inner product[·,·] and norm · and let P be a continuous map from X to itself such that
P (ζ ), ζ>0 forζ = k > 0. (2.13)
Then there exists ζ∈ X with ζ k so that P (ζ ) = 0.
In applying Lemma 4, we take X= the space spanned by w1, . . . , wnand the inner product
[·,·] is induced by that of V , namely, ·,· given in (2.7). Here the norm · = · H1(Ω). We
Pn(v), w =Pn(v), w = μv, w + εb(v0, v, w)+ εb(v, v0, w)+ ε2b(v, v, w)− (f, w)
for v, w∈ X. The continuity of Pnis obvious. To verify (2.13), we can see with the help of (2.10),
(2.11), and (2.12) that Pn(v), v = μv, v + εb(v0, v, v)+ εb(v, v0, v)+ ε2b(v, v, v)− (f, v) Cv2 H1(Ω)−εb(v, v, v0) −(f, v) Cv2 H1(Ω)− |ε|C v0H2(Ω)v2 H1(Ω)− f H−1(Ω)vH1(Ω) vH1(Ω) C− |ε|C v0H2(Ω) vH1(Ω)− f H−1(Ω) ,
where C and C are positive numbers. Therefore, if we choose a small ε0, depending on ψ , such
that
C− |ε|C v0H2(Ω)>0 ∀|ε| ε0,
then[Pn(v), v] > 0 for vH1(Ω)= k with
k > f H−1(Ω) C− |ε|C v0H2(Ω)
for|ε| ε0.
Hence, Lemma 4 guarantees the existence of vnsatisfying (2.8) and (2.9).
Now we want to pass the limit of vn. Multiplying (2.9) by ¯ξj,nand summing the corresponding
equalities from 1 to n gives
μvn, vn + εb(v0, vn, vn)+ εb(vn, v0, vn)+ ε2b(vn, vn, vn)= (f, vn).
Using (2.10), (2.11), and (2.12) again, we obtain that
vnH1(Ω) C0f H−1(Ω),
where C0>0 is uniformly in ε provided|ε| ε0. Therefore, there exist v in V and a subsequence
{vn } such that
vn → v weakly in V. (2.14)
By the Rellich theorem, we have that
vn → v strongly in L2(Ω). (2.15)
With the help of (2.14) and (2.15), we can derive that
b(v0, vn , w)→ b(v0, v, w), (2.16) b(vn , v0, w)→ b(v, v0, w), (2.17)
and
b(vn , vn , w)→ b(v, v, w) (2.18)
for all w∈ V. The proof of (2.18) was already given in [23, Lemma 1.5, Chapter II]. As for (2.16) and (2.17), they are easy consequences of (2.15). Therefore, passing the limit in vn , we obtain
that
μv, wj + εb(v0, v, wj)+ εb(v, v0, wj)+ ε2b(v, v, wj)= (f, wj)
for j= 1, 2, . . . . Since w1, w2, . . .is a complete orthonormal system of V , we conclude that μv, w + εb(v0, v, w)+ εb(v, v0, w)+ ε2b(v, v, w)= (f, w)
for all w∈ V . Thus, there exists q ∈ L2(Ω)(and v∈ V ) such that − divσμ(v, q)
+ ε(v0· ∇)v + ε(v · ∇)v0+ ε2(v· ∇)v = f
in the weak sense. 2
Now we have the existence of weak solution (v, q) to (2.4). To indicate the dependence of
(v, q)on ε, we denote v= ˜vε and q= ˜qε. Our next task is to derive the regularity of (˜vε,˜qε).
The aim is to show that (˜vε,˜qε)are uniformly bounded in ε with respect to some Sobolev norms.
This enables us to consider the limiting behavior of (ε−1vε, ε−1pε). The proof of regularity for
(˜vε,˜qε)relies on the regularity result for the Stokes equations and the “bootstrapping” technique.
Some arguments used here are inspired by [23].
Theorem 5. Let (˜vε,˜qε) be a weak solution of (2.4) for|ε| ε0. We may choose ε0<1. Then
(˜vε,˜qε)∈ H2(Ω)× H1(Ω) and satisfies ˜vεH2(Ω)+ ˜qεH1(Ω)/R C 16 j=2 ψj H3/2(∂Ω) (2.19) where C is independent of ε.
Proof. To simply the notations in the proof, we reuse v= ˜vεand q= ˜qε. We now write the first
equation of (2.4) in the form of Stokes equations divσμ(v, q)
= g, div v = 0 with
g= ε(v0· ∇)v + ε(v · ∇)v0+ ε2(v· ∇)v − f.
From the proof of the existence, we see that v∈ H01(Ω)and
Hereafter, C, C , C , and ˜C represent constants independent of ε. Their exact values are not important. In view of the Sobolev imbedding, we have v∈ L6(Ω). Subsequently, we get that
(v· ∇)v ∈ L3/2(Ω)and
(v· ∇)v L3/2(Ω) Cv2H1(Ω) Cψ
4
H3/2(∂Ω).
Similarly, we have that
(v· ∇)v0 L3/2(Ω) CvL6(Ω)v0H1(Ω) Cψ4H3/2(∂Ω)
and
(v0· ∇)v L3/2(Ω) C (v0· ∇)v L2(Ω) Cψ4H3/2(∂Ω).
Therefore, from (2.6) we have g∈ L3/2(Ω)and gL3/2(Ω) C
ε0ψ4H3/2(∂Ω)+ ψ2H3/2(∂Ω)
.
From now on, we choose ε0<1 and hence
gL3/2(Ω) C ψ4 H3/2(∂Ω)+ ψ 2 H3/2(∂Ω) .
The regularity theorem for the Stokes equations (Theorem 11 in Appendix A) implies vW2,3/2(Ω)+ qW1,3/2(Ω)/R CgL3/2(Ω) C ψ4 H3/2(∂Ω)+ ψ 2 H3/2(∂Ω) . (2.20)
The estimate (2.20) is not exactly what we want. We need to improve L3/2-base Sobolev norms to L2-base ones on the left-hand side of (2.20). This can be achieved by the “bootstrap-ping” argument. In view of Sobolev imbedding, W2,3/2(Ω) → Lr(Ω)for any 1 < r <∞, we thus obtain that v⊗ v ∈ Lsfor any s∈ (1, ∞) and
(v· ∇)v W−1,s(Ω)= ∇(v⊗ v) W−1,s(Ω) CvLr1(Ω)vLr2(Ω) C v2 W2,3/2(Ω) C ψ8 H3/2(∂Ω)+ ψ4H3/2(∂Ω) , (2.21)
where s−1= r1−1+ r2−1 and C = C (Ω, r1, r2). In the first equality of (2.21), we have used div v= 0. On the other hand, v0∈ H2(Ω)= W2,2(Ω) → W2,3/2(Ω). Likewise, we have that
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ (v· ∇)v0 W−1,s(Ω) ˜C ψ5 H3/2(∂Ω)+ ψ 3 H3/2(∂Ω) , (v0· ∇)v W−1,s(Ω) ˜Cψ5 H3/2(∂Ω)+ ψ 3 H3/2(∂Ω) , f W−1,s(Ω)= (v0· ∇)v0 W−1,s(Ω) ˜Cψ 2 H3/2(∂Ω), (2.22)
where s−1= r1−1+ r2−1and ˜C= ˜C(Ω, r1, r2). Combining (2.21) and (2.22) leads to gW−1,s(Ω) C 8 j=2 ψj H3/2(∂Ω).
Thus from Theorem 10 in Appendix A we have
vW1,s(Ω)+ qLs(Ω)/R C 8 j=2 ψj H3/2(∂Ω) for any s∈ (1, ∞).
We need one more iteration. Due to the imbedding theorem, i.e., W1,s(Ω) → C0( ¯Ω) for
s >3, we can see that (v· ∇)v ∈ Ls(Ω)for any s∈ (1, ∞) since Ω is bounded. In particular, we have (v· ∇)v ∈ L2(Ω)and (v· ∇)v L2(Ω) C 16 j=4 ψj H3/2(∂Ω).
We already knew that v0∈ W2,2(Ω) → C0( ¯Ω). So we immediately have
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (v· ∇)v0 L2(Ω) C 9 j=3 ψj H3/2(∂Ω), (v0· ∇)v L2(Ω) C 9 j=3 ψj H3/2(∂Ω), f L2(Ω) Cψ2 H3/2(∂Ω).
In other words, we get
gL2(Ω) C 16 j=2 ψj H3/2(∂Ω)
and therefore Theorem 11 in Appendix A implies
vH2(Ω)+ qH1(Ω)/R C 16 j=2 ψj H3/2(∂Ω). 2
Remark 6. Even though we will not need the uniqueness result of direct problem (2.1) in our
inverse problem, the solution (uε, pε)of (2.1) with boundary condition φ= εψ is indeed unique
(pεis unique up to constants) when|ε| is sufficiently small. This fact can be proved by modifying
3. Inverse problem
We already proved the existence of the solution to the system (1.1) with the boundary data
εψin Section 2. More importantly, we derived the asymptotic behavior of solutions as ε→ 0. In this section we linearize the Cauchy data ˜Sμand prove Theorems 1 and 2.
Given any ψ∈ H3/2(∂Ω), let (v0, q0)∈ H2(Ω)×H1(Ω)be the unique solution (q0is unique
up to a constant) of the Stokes equations divσμ(v0, q0) = 0 in Ω, div v0= 0 in Ω (3.1)
with boundary data v0|∂Ω= ψ (see Theorem 11 in Appendix A). As proved in the previous
section, there exists (uε, pε)with the form
uε= εv0+ ε2˜vε, pε= εq0+ ε2˜qε
satisfying (1.1) with boundary data uε|∂Ω = εψ for all |ε| < ε0, where ε0 depends on
ψH3/2(∂Ω). Moreover, ˜vεand ˜qεsatisfy (2.19). We immediately see that
ε−1uε− v0 H2(Ω)= ε ˜vεH2(Ω)→ 0, ε−1pε− q0 H1(Ω)/R= ε ˜qεH1(Ω)/R→ 0, and therefore ε−1uε|∂Ω− v0|∂Ω H3/2(∂Ω)→ 0, (3.2) ε−1σμ(uε, pε)n|∂Ω− σμ(v0, q0)n|∂Ω H1/2(∂Ω)→ 0 (3.3) provided Ω pεdx= Ω q0dx= 0.
As in [7], we define the Cauchy data associated to (3.1)
Sμ=
v0|∂Ω, σμ(v0, q0)n|∂Ω
⊂ H3/2(∂Ω)× H1/2(∂Ω), (3.4)
where (v0, q0)satisfies (3.1). Now combining (3.2) and (3.3) leads to the following result:
Theorem 7. The Cauchy data Sμof the Stokes equations can be uniquely determined from the
Cauchy data ˜Sμof the Navier–Stokes equations.
In other words, let μ1and μ2be two viscosities, then ˜Sμ1= ˜Sμ2 implies Sμ1 = Sμ2. So we
reduce the uniqueness question of the inverse problem for the Navier–Stokes equations to that for the Stokes equations. Therefore, Theorems 1 and 2 follow from Theorem 7 and the unique determination of viscosity for the Stokes equations proved in [7].
Theorem 8. (See [7, Theorem 1.1].) Assume that μ1(x) and μ2(x) are two viscosity functions satisfying μ1, μ2∈ Cn0( ¯Ω) for n0 8 and
∂αμ1(x)= ∂αμ2(x) ∀x ∈ ∂Ω, |α| 1.
Let Sμ1 and Sμ2 be the Cauchy data associated with μ1and μ2, respectively. If Sμ1= Sμ2 then
μ1= μ2.
Theorem 9. (See [7, Corollary 1.4].) Let ∂Ω be convex with nonvanishing Gauss curvature.
Assume that μ1(x) and μ2(x) are two viscosity functions satisfying μ1, μ2∈ Cn0( ¯Ω) for n0 8. If Sμ1= Sμ2 then μ1= μ2.
The regularity requirement in Theorem 8 is to make sure that Eskin’s method of [3] works in our case. We do not know whether it is optimal. Also, we want to remark that in [7] we define the Cauchy data of the Stokes equations as a subset of H1/2(∂Ω)× H−1/2(∂Ω), i.e.
Sμ⊂ H1/2(∂Ω)× H−1/2(∂Ω). Nevertheless, the same proof in [7] still holds true when we
consider Sμas given in (3.4).
Acknowledgment
The second author would like to thank Professor Tai-Peng Tsai at UBC for some helpful discussions.
Appendix A. Ls(s > 1) theory for the Stokes equations
In this appendix we will prove the existence, uniqueness, and regularity of the solution to the Stokes equations in the category of Ls when the viscosity is a function. When μ is a constant, this problem has been well documented in the literature, see for example [6,23]. However, we were not able to find any reference for the case where μ is a variable function. For the sake of completeness, we provide a proof here. As before, let Ω be an open bounded domain inR3with smooth boundary ∂Ω. Consider the Stokes equations
⎧ ⎨ ⎩ div σμ(v, q)= f in Ω, div v= g in Ω, v= ψ on ∂Ω, (A.1)
where the viscosity μ(x) > 0 and the following compatibility condition holds: Ω g dx= ∂Ω ψ· n ds.
The system (A.1) will not be affected if we add a constant to q. We define the following norm qWk,s(Ω)/R:= inf
c∈Rq + cW
For a suitable μ(x), one can prove that
Theorem 10. Suppose μ∈ C1,1( ¯Ω). For any
f ∈ W−1,s(Ω), g∈ Ls(Ω), ψ∈ W1−1/s,s(∂Ω), 1 < s <∞,
there exists a unique solution (v, q)∈ W1,s(Ω)×Ls(Ω)(q is unique up to a constant) satisfying (A.1). Moreover this solution obeys the estimate
vW1,s(Ω)+ qLs(Ω)/R
Cf W−1,s(Ω)+ gLs(Ω)+ ψ
W1−1/s,s(∂Ω)
(A.2)
where C depends on s, Ω , minx∈ ¯Ωμ andμC1,1( ¯Ω).
Theorem 11. Suppose μ∈ Cr,1( ¯Ω) with the integer r= max{m, 1}, m 0. For any f ∈ Wm,s(Ω), g∈ Wm+1,s(Ω), ψ∈ Wm+2−1/s,s(∂Ω), 1 < s <∞,
there exists a unique solution (v, q)∈ Wm+2,s(Ω)× Wm+1,s(Ω)(q is unique up to a constant)
satisfying (A.1). Moreover this solution obeys the estimate
vWm+2,s(Ω)+ qWm+1,s(Ω)/R
Cf Wm,s(Ω)+ gWm+1,s(Ω)+ ψWm+2−1/s,s(∂Ω)
(A.3)
where C depends on m, s, Ω, minx∈ ¯Ωμ andμCr,1( ¯Ω).
Remark 12. In Theorems 10 and 11, the regularity assumptions on μ are not necessarily
op-timal. We impose the least smoothness μ∈ C1,1( ¯Ω)due to the consideration of uniqueness in Lemma 14.
When μ is a constant, these results were originally proved by Cattabriga [2], and a nice presen-tation could be found for example in [6, Theorem IV.6.1 and Ex. IV.6.2]. Here we shall provide a proof for the variable viscosity following general procedures used in [2,6]. From [6, Theo-rem III.3.2], we know there exists at least one vector field w∈ Wm+2,s(Ω)(m −1) such that
div w= g in Ω,
w= ψ on ∂Ω. Moreover, this solution satisfies the estimate
wWm+2,s(Ω) C(m, s, Ω)
gWm+1,s(Ω)+ ψWm+2−1/s,s(∂Ω)
.
So we can always assume that g and ψ are zeros in system (A.1). Theorems 10 and 11 will be proved together. We first show a priori estimates.
Lemma 13. Let (v, q)∈ W1,s(Ω)× Ls(Ω) be a weak solution of (A.1).
(i) Suppose μ∈ C0( ¯Ω), f , g, and ψ are as in Theorem10. We have vW1,s(Ω)+ qLs(Ω)/R
Cf W−1,s(Ω)+ gLs(Ω)+ ψW1−1/s,s(∂Ω)+ vLs(Ω)+ qW−1,s(Ω). (A.4)
(ii) Suppose μ∈ Cm,1( ¯Ω) (m 0), f , g, and ψ are as in Theorem 11. We have
vWm+2,s(Ω)+ qWm+1,s(Ω)/R
Cf Wm,s(Ω)+ gWm+1,s(Ω)+ ψWm+2−1/s,s(∂Ω)+ vLs(Ω)+ qW−1,s(Ω). (A.5)
Proof. We only need to consider the case when g and ψ are zeros. The estimate (A.4) can be
proved by freezing the coefficient and the result for the Stokes equations with constant vis-cosity. One only needs to show (A.4) holds in a ball BR(x(0))and a half ball BR+(x(1)):=
BR(x(1))∩{x3>0} which correspond to the interior estimate and the estimate near the boundary,
respectively. Then a finite covering of ¯Ωimplies that (A.4) holds. Consider a ball B2R(x(0))⊂ Ω
and take a cutoff function η∈ C0∞(B2R(x(0)))with η= 1 in BR(x(0)). Direct computation shows
that ηv and ηq satisfy the following constant coefficient Stokes equations ⎧ ⎪ ⎨ ⎪ ⎩ μx(0) (ηv)− ∇(ηq) = ˜f in B2R x(0), div(ηv)= ˜g in B2R x(0), ηv= 0 on ∂B2R x(0), (A.6) where ˜ f= − div2μ(x)− μx(0)Sym∇(ηv)+ ηf
+ div2μ(x)− μx(0)Sym(∇η ⊗ v)+ 2μ(x)− μx(0)∇η · Sym(∇v)
+ 2μx(0)(∇η · ∇) · v + μx(0)( η)v+ q∇η
and˜g = ∇η · v.
From [6, Theorem IV.6.1 and Ex. IV.6.2], we have ηvW1,s(B 2R(x(0)))+ ηqLs(B2R(x(0)))/R C1 ˜fW−1,s(B2R(x(0)))+ ˜gLs(B2R(x(0))) C2 sup x∈B2R(x(0)) μ(x)− μx(0)ηvW1,s(B 2Rl(x(0))) + C3 f W−1,s(B2R(x(0)))+ vLs(B2R(x(0)))+ qW−1,s(B2R(x(0))) .
If the radius 2R of the ball is small, then supx∈B2R(x(0))|μ(x)−μ(x
(0))| will be small too because
of the continuity of μ. So the termηvW1,s(B
2R(x(0))) can be absorbed to the left-hand side.
vW1,s(B
R(x(0)))+ qLs(BR(x(0)))/R
Cf W−1,s(B2R(x(0)))+ vLs(B2R(x(0)))+ qW−1,s(B2R(x(0)))
.
A similar argument works for the half ball. Then (A.4) holds by a finite covering of ¯Ω.
The estimate (A.5) can be proved by the iteration for m > 0 provided it is true for m= 0. If
(v, q)∈ W2,s(Ω)× W1,s(Ω), then (A.5) holds for m= 0 by a similar proof as for (A.4). We use the difference quotient method to show (v, q)∈ W2,s(Ω)× W1,s(Ω). The definition of the difference quotient of a function w∈ Wk,s(Ω) (k 0) in the direction ej is given by
wh(x)=w(x+ hej)− w(x)
h , h= 0.
A useful estimate is
wh
Wk−1,s(Ω ) CwWk,s(Ω) (A.7)
for any Ω satisfying ¯Ω ⊂ Ω and |h| < dist(Ω , ∂Ω).
As before, we consider the case for a ball and a half ball only. Direct computation shows that
(ηv)h and (ηq)h satisfy the Stokes equations (A.6) with the data ˜fh, ˜gh. The first term in ˜fh
will bother us a little bit. We compute div2μ(x)− μx(0)Sym∇(ηv)h = div2μ(x)− μx(0)Sym∇(ηv)h = div2(μh(x)Sym∇(ηv)(x + hej) + div2μ(x)− μx(0)Sym∇(ηv)h.
From the estimate (A.4), (A.7), letting R be small as before, we know (ηv)h W1,s(B
2R(x(0)))+
(ηq)h Ls(B
2R(x(0)))/R C.
So (v, q)∈ W2,s(BR(x(0)))× W1,s(BR(x(0))).
For a half ball BR+(x(1)):= BR(x(1))∩ {x3>0}, using the same argument as for a ball, we
can conclude that the derivatives DjDkv∈ L2(BR+(x(1)))for all pairs (j, k)= (3, 3), and Dj ∈
L2(BR+(x(1)))for j= 3. Then using this obtained result and the system (A.1) satisfied by (v, q), we can get D3D3v∈ L2(BR+(x(1)))and D3∈ L2(BR+(x(1))). The proof is complete. 2
We are going to drop the last two terms in (A.4) and in (A.5). We need the following unique-ness result.
Lemma 14. Suppose that μ∈ C1,1( ¯Ω). If (v, q)∈ W1,s(Ω)× Ls(Ω) is a weak solution of(A.1)
with f= g = 0 in Ω and ψ = 0 on ∂Ω, then v = 0, q = const a.e. in Ω.
Proof. If s= 2, the uniqueness of the solution is proved in Section 2 of [7]. If s > 2, it is also
m= 1, we know (v, q) ∈ W3,s(Ω)× W2,s(Ω). Then the uniqueness result for 1 < s < 2 follows from the Sobolev imbedding
W3,s(Ω)× W2,s(Ω)⊂ W1,2(Ω)× L2(Ω), for any 1 < s < 2, and the uniqueness result for s= 2. 2
Arguing as in [6, Lemma IV.6.1], one can drop the last two terms in (A.4) and in (A.5) by Lemma 14 and compactness properties of Sobolev spaces. We obtain
Lemma 15. Let (v, q)∈ W1,s(Ω)× Ls(Ω) be a weak solution of (A.1). (i) Suppose that μ, f , g and ψ are as in Theorem 10. We have
vW1,s(Ω)+ qLs(Ω)/R
Cf W−1,s(Ω)+ gLs(Ω)+ ψW1−1/s,s(∂Ω)
. (A.8)
(ii) Suppose that μ, f , g and ψ are as in Theorem 11. We have vWm+2,s(Ω)+ qWm+1,s(Ω)/R
Cf Wm,s(Ω)+ gWm+1,s(Ω)+ ψWm+2−1/s,s(∂Ω)
. (A.9) To finish the proofs for Theorems 10 and 11, one only needs to show the existence of weak solution. In view of (A.9), we only need to prove Theorem 10.
Lemma 16. Suppose that μ, f , g and ψ are as in Theorem 10. Then there exists a weak solution
(v, q)∈ W1,s(Ω)× Ls(Ω) of (A.1).
Proof. Once again, we only consider the case while g and ψ are zeros. When s= 2, the existence
of weak solution is proved in Section 2 of [7]. For a general s∈ (1, ∞), we argue as follows. We first prove that the weak solution (v, q)∈ W1,s(Ω)× Ls(Ω) exists when f ∈ W1,2(Ω).
Since f ∈ W1,2(Ω)⊂ W−1,2(Ω), we already showed the existence of weak solution (v, q)∈
W1,2(Ω)× L2(Ω). Moreover, from (A.9) of Lemma 15 with m= 1, s = 2 and an imbedding theorem, we obtain
(v, q)∈ W3,2(Ω)× W2,2(Ω)⊂ W1,s(Ω)× Ls(Ω).
Now we consider the case when f ∈ W−1,s(Ω). Let us choose a sequence {fj} ⊂ C0∞⊂
W1,2(Ω) such that fj − f W−1,s(Ω) → 0 as j → ∞. Then there exist weak solutions
(vj, qj)∈ W1,s(Ω)× Ls(Ω)corresponding to the data fj. Using (A.8) we get that there
ex-ists (v, q)∈ W1,s(Ω)× Ls(Ω)such that
vj→ v strongly in W1,s(Ω), qj→ q strongly in Ls(Ω).
References
[1] A.P. Calderón, On an inverse boundary value problem, in: Seminar on Numerical Analysis and Its Applications to Continuum Physics, Soc. Brasileira de Mathemática, Rio de Janeiro, 1980, pp. 65–73.
[2] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Univ. Padova 31 (1961) 308–340.
[3] G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang–Mills potentials, Comm. Math. Phys. 222 (2001) 503–531.
[4] G. Eskin, J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems 18 (2002) 907–921.
[5] G. Eskin, J. Ralston, Inverse boundary value problems for systems of partial differential equations, in: Recent Development in Theories & Numerics, World Sci. Publishing, River Edge, NJ, 2003, pp. 105–113.
[6] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vols. I, II, Springer-Verlag, 1994.
[7] H. Heck, X. Li, J.-N. Wang, Identification of viscosity in an incompressible fluid, Indiana Univ. Math. J., in press. [8] D. Hervas, Z. Sun, An inverse boundary value problem for quasilinear elliptic equations, Comm. Partial Differential
Equations 27 (2002) 2449–2490.
[9] V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Ration. Mech. Anal. 124 (1993) 1–13.
[10] V. Isakov, Uniqueness of recovery of some quasilinear partial differential equations, Comm. Partial Differential Equations 26 (2001) 1947–1973.
[11] V. Isakov, Uniqueness of recovery of some systems of semilinear partial differential equations, Inverse Problems 17 (2001) 607–618.
[12] V. Isakov, A. Nachman, Global uniqueness in a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc. 347 (1995) 3375–3390.
[13] V. Isakov, J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math. 47 (1994) 1403–1410.
[14] H. Kang, G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems 18 (2002) 1079–1088.
[15] G. Nakamura, Z. Sun, An inverse boundary value problem for St. Venant–Kirchhoff materials, Inverse Problems 10 (1994) 1159–1163.
[16] G. Nakamura, G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math. 118 (1994) 457–474.
[17] G. Nakamura, G. Uhlmann, Erratum: “Global uniqueness for an inverse boundary value problem arising in elastici-ty” (Invent. Math. 118 (1) (1994) 457–474), Invent. Math. 152 (2003) 205–207.
[18] Z. Sun, On a quasilinear inverse boundary value problem, Math. Z. 221 (1996) 293–305.
[19] Z. Sun, Inverse boundary value problems for a class of semilinear elliptic equations, Adv. in Appl. Math. 32 (2004) 791–800.
[20] Z. Sun, Conjectures in inverse boundary value problems for quasilinear elliptic equations, Cubo 7 (2005) 65–73. [21] Z. Sun, G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math. 119 (1997) 771–797. [22] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.
(2) 125 (1987) 153–169.
[23] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1977. [24] G. Uhlmann, Developments in inverse problems since Calderón’s foundational paper, in: Harmonic Analysis and
Partial Differential Equations (Essays in Honor of Alberto P. Calderón), The University of Chicago Press, Chicago, 1999, pp. 295–345.
