# Inverse Boundary Value Problem for the Magnetohydrodynamics Equations

Authors: Lianhua He; Zhong Tan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider the stationary magnetohydrodynamics (MHD) equations in a bounded domain ofℝdd=2,3 with viscosity and magnetic diffusing. By the linearization technique, we prove that the uniqueness of viscosity function and magnetic diffusing function in the MHD equations is determined from the knowledge of the Cauchy data measured on the boundary.


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## Body

## 1. Introduction

LetΩ⊂ℝdd=2,3 be an open bounded domain with boundary ∂Ω∈C∞. Assume that Ω is filled with an incompressible fluid. u∈ℝd and B∈ℝd stand for velocity field and magnetic field, respectively. u,B satisfies the following stationary magnetohydrodynamics(MHD) equations:(1)divσμu,p−u·∇u+B·∇B=0inΩ,divu=0inΩ,divσ~λB−u·∇B+B·∇u=0inΩ,divB=0inΩ.where(2)σμu,p=2μSym∇u−pI;σ~λB=2λSym∇B,and SymA=A+AT/2 is the symmetric part of the matrix A. p is the pressure. The notation I is the identity matrix. Here, μx>0 is the viscosity function and λx>0 is the magnetic diffusivity function.In this paper, we are interested in the inverse problem for MHD equation. First, we define the Cauchy data for the MHD equation (1) by(3)C~μ,λ=u∂Ω,B∂Ω,σμu,pn∂Ω,σ~λBn∂Ωu,p,Bsatisfiestheequations,where n is the unit outer normal of the boundary ∂Ω and u∂Ω, and B∂Ω satisfies the compatibility conditions(4)∫∂Ωu∂Ω·nds=0,∫∂ΩB∂Ω·nds=0.In the physical sense,σμu,pn∂Ω stands for the stress acting on ∂Ω and is called the Cauchy force. The motivation of this paper is to determine μ,λ from the knowledge of the Cauchy data C~μ,λ.To discuss the inverse problem, we will not consider the general Dirichlet datau,B∂Ω=ϕ1,ϕ2. Particularly, we shall assume(5)ϕ1,ϕ2=εψ1,εψ2,with ∣ε∣ sufficiently small and ψi∈H3/2∂Ωi=1,2 satisfying the compatibility condition (4). For such a choice of Dirichlet data, we can obtain that there exists a solution u,p,B of the equation (1) with u∂Ω=εψ1, B∂Ω=εψ2 and the boundary trace σμu,pn∂Ω,σ~λBn∂Ω∈H1/2∂Ω. Thus, the Cauchy data C~μ,λ is meaningful in this case. When ∣ε∣ is sufficiently small, we even know that the solution u,p,B to (1) is unique (p is unique up to a constant), but we do not need it. The main results of this paper are the following global uniqueness theorems of the inverse problem.Theorem 1.
LetΩ⊂ℝ3. Assume that μ1x and μ2x are viscosity functions and λ1x and λ2x are magnetic diffusivity functions. μi,λi∈Cn0Ω¯i=1,2 for n0≥8 and(6)∂αμ1x=∂αμ2x∀x∈∂Ω,α≤1,(7)∂βλ1x=∂βλ2x∀x∈∂Ω,∣β∣≤1.
LetC~μ1,λ1 and C~μ2,λ2 be the Cauchy data with μ1,λ1 and μ2,λ2, respectively. If C~μ1,λ1=C~μ2,λ2, then μ1,λ1=μ2,λ2.
When the boundary∂Ω is convex and has nonvanishing Gauss curvature, we can remove the assumption (6)(7) from Theorem 1.Theorem 2.
LetΩ⊂ℝ3. Suppose that ∂Ω is convex with the nonvanishing Gauss curvature, and μix and λixi=1,2 are viscosity functions and magnetic diffusivity functions, respectively, satisfying μi,λi∈Cn0Ω¯ for n0≥8. If C~μ1,λ1=C~μ2,λ2, then μ1,λ1=μ2,λ2.Theorem 3.
LetΩ be a simply connected bounded domain in ℝ2 with smooth boundary. Suppose that μix and λixi=1,2 are viscosity functions and magnetic diffusivity functions, respectively. Assume that μix,λix∈C3Ω¯ and μix,λix>0 with(8)∂αμ1∂Ω=∂αμ2∂Ωforallα≤1,∂βλ1∂Ω=∂βλ2∂Ωforall∣β∣≤1.
LetC~μ1,λ1 and C~μ2,λ2 be the Cauchy data associated with μ1,λ1 and μ2,λ2, respectively. If C~μ1,λ1=C~μ2,λ2, then μ1,λ1=μ2,λ2.If we attempt to determine the internal parameters of body, we can make measurements only at the surface of the body. This is the well-known inverse boundary value problems. A typical application of the inverse problem is electrical impedance tomography (EIT). Since the 1980s, the parameter determination problem by boundary measurements has been well studied. Since Calderón’s pioneer contribution [1], the method of complex geometrical optics solutions (see [2–4]) which was introduced by Sylvester and Uhlmann [5] has become a standard method. There are other methods to research the inverse boundary value problems, for example, by using the Dirichlet-to-Neumann map (see [6–8]), complex exponential solution (see [9–11]), and Cauchy data ([12, 13]). The global uniqueness of identifying the viscosity using the Cauchy data is a rather well-studied field. For the Stokes equation, the uniqueness for the inverse boundary problem was considered by Heck et al. [9] and Lai et al. [12]. The unique determination of the viscosity μ for the Navier-Stokes equations is proved by Li and Wang [14] in dimension three, Imanuvilov and Yamamoto [7] in dimension two, and Lai et al. [12] in dimension two. In [12, 14–16], they used the linearization method to study the uniqueness determination of μ for the Navier-Stokes equations. This method was first introduced by Isakov [17] in a semilinear parabolic inverse problem. This technique allows for the reduction of the semilinear inverse boundary value problem to the corresponding linear one.In this paper, we would like to apply the linearization technique to consider the unique determination problem of parameters in MHD equations. The main difficulty in applying the linearization technique to solve our problem lies in the existence of particular solutions to (1) which has some controlled asymptotic properties. In order to solve this difficulty, we only consider the Dirichlet condition with small parameter ε as in (5). The key step in the proofs of Theorems 1, 2, and 3 is to prove the existence of the solution uε,pε,Bε to (1) with boundary condition (5) and ε−1uε,ε−1pε,ε−1Bε converging to u0,p0,B0 in suitable Sobolev spaces, where u0,p0 and B0 satisfy two Stokes equations, respectively, whereafter the Cauchy data C~μ,λ of MHD equation can determine Cauchy data Cμ of the Stokes equation about u and Cλ of the Stokes equation about B. Then, the inverse problem for the MHD equation (1) is reduced to the same problem for the Stokes equation. The innovation point in this paper is the reduction of the nonlinear inverse boundary value problem to two corresponding ones.This paper is organized as follows. In Section2, we will prove the existence of the boundary value problem for (1). In Section 3, we linearize the Cauchy data C~μ,λ and prove Theorems 1, 2, and 3.


## 2. Direct Problem

In this section, we would like to prove the existence of the boundary value problem:(9)divσμu,p−u·∇u+B·∇B=0inΩ,divσ~λB−u·∇B+B·∇u=0inΩ,divu=0,divB=0inΩ,u=ϕ1,B=ϕ2on∂Ω,with ϕ_ii=1,2∈H3/2∂Ω and the compatibility conditions (4). When μ and λ are constants, this problem has been discussed in literature [18].In order to prove the existence of equation (9), we first introduce some lemmas.Lemma 4 (see [14], Theorem 10).
Supposeμ∈C1,1Ω¯. For any,(10)f∈W−1,sΩ,g∈LsΩ,ψ∈W1−1/s,s∂Ω,1<s<∞,there exists a unique solution u,p∈W1,sΩ×LsΩ (p is unique up to a constant) satisfying(11)divσμu,p=finΩ,divu=ginΩ,u=ψon∂Ω.
Moreover, this solution obeys the estimate(12)∥u∥W1,sΩ+∥p∥LsΩ/ℝ≤C∥f∥W−1,sΩ+∥g∥LsΩ+∥ψ∥W1−1/s,s∂Ω,where C depends on s,Ω,minx∈Ω¯μ and ∥μ∥C1,1Ω¯.Lemma 5 (see [14], Theorem 11).
Supposeμ∈Cr,1Ω¯ with the integer r=maxm,1, m≥0. For any,(13)f∈Wm,sΩ,g∈Wm+1,sΩ,ψ∈Wm+2−1/s,s∂Ω,1<s<∞,there exists a unique solution u,p∈Wm+2,sΩ×Wm+1,sΩ (p is unique up to a constant) satisfying (11). Moreover, this solution obeys the estimate(14)∥u∥Wm+2,sΩ+∥p∥Wm+1,sΩ/ℝ≤C∥f∥Wm,sΩ+∥g∥Wm+1,sΩ+∥ψ∥Wm+2−1/s,s∂Ω,where C depends on m,s,Ω, minx∈Ω¯μ, and ∥μ∥Cr,1Ω¯.Lemma 6 (see [19], Lemma 1.4, Chapter II).
LetX be a finite-dimensional Hilbert space with inner product ·,· and norm ∥·∥. And let P be a continuous map from X to itself such that(15)Pζ,ζ>0for∥ζ∥=k>0.
Then, there existsζ∈X with ∥ζ∥≤k so that Pζ=0.
As presented in the introduction, we assumeϕi=εψii=1,2 with ψi∈H3/2∂Ω and look for uε,pε,Bε=εuε,εpε,εBε satisfies (9). Then, the problem (9) is reduced to(16)divσμuε,pε−εuε·∇uε+εBε·∇Bε=0inΩ,divσ~λBε−εuε·∇Bε+εBε·∇uε=0inΩ,divuε=0,divBε=0inΩ,uε=ψ1,Bε=ψ2on∂Ω.
We will prove the problem (16) with the solution uε=u0+εu, pε=p0+εp, Bε=B0+εB, where u0,p0,B0 satisfies the Stokes equation(17)divσμu0,p0=0inΩ,divu0=0inΩ,u0=ψ1on∂Ω,(18)divσ¯λB0=0inΩ,divB0=0inΩ,B0=ψ2on∂Ω.
u,p,B satisfies(19)−divσμu,p+εu0·∇u+εu·∇u0+ε2u·∇u−εB0·∇B−εB·∇B0−ε2B·∇B=B0·∇B0−u0·∇u0≜f1inΩ,−divσ~λB−εB0·∇u−εB·∇u0−ε2B·∇u+εu·∇B0+εu0·∇B+ε2u·∇B=−u0·∇B0+B0·∇u0≜f2inΩ,divu=0,divB=0inΩ,u=0,B=0on∂Ω.
By Lemma5, we can know that for each ψ1,ψ2∈H3/2∂Ω there exists a unique u0,p0,B0∈H2Ω×H1Ω×H2Ω (p0 is unique up to a constant) satisfying (17) and (18) and the estimate(20)∥u0∥H2Ω+∥p0∥H1Ω/ℝ≤C∥ψ1∥H3/2∂Ω;∥B0∥H2Ω≤C∥ψ2∥H3/2∂Ω,with ∥p0∥H1Ω/ℝ≔infc∈ℝ∥p0+c∥H1Ω. In view of the Sobolev imbedding theorems H2Ω↪C0Ω¯ and H1Ω↪L4Ω, we get that(21)∥f1∥H1Ω=∥u0·∇u0−B0·∇B0∥H1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2,(22)∥f2∥H1Ω=∥B0·∇u0−u0·∇B0∥H1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2.
In order to solve the problem (19), we first prove the existence of (19).Theorem 7.
There exists a positive numberε0 depending on ψ1,ψ2 such that for any ∣ε∣≤ε0, problem (19) has at least one weak solution u,p,B∈H01Ω×L2Ω×H01Ω.Proof.
Similar to Chapter II, Section1 in [19], we use the Galerkin method to solve the problem (19). We denote V=u∈H01Ω;divu=0, V=V×V. ∥u,B∥V=∥u∥H1Ω2+∥B∥H1Ω21/2. By Korn’s inequality and Poincare’s inequality, it is easily to prove that H01Ω is a separable Hilbert space with respect to the inner product(23)<u,w>=∫ΩSu·Sw¯dx.
Note thatV is a closed subspace of H01Ω, which is also separable. Let ω1,ω2,⋯ be elements of V which form a complete orthonormal system of V, where(24)V≔w∈C0∞Ω,divw=0.
Letun=∑j=1nξj,nωj, Bn=∑j=1nζj,nωj with ξj,n,ζj,n∈ℂ satisfy(25)μ<un,ωj>+εbu0,un,ωj+εbun,u0,ωj+ε2bun,un,ωj,(26)−εbB0,Bn,ωj−εbBn,B0,ωj−ε2bBn,Bn,ωj=f1,ωj(27)λ<Bn,ωj>−εbB0,un,ωj−εbBn,u0,ωj−ε2bBn,un,ωj,(28)+εbun,B0,ωj+εbu0,Bn,ωj+ε2bun,Bn,ωj=f2,ωj,for j=1,2,⋯,n, where(29)bu,v,w=∫Ωu·∇v·w¯dx,f,w=∫Ωf·w¯dx.
Analogous to Lemma 1.3, Chapter II in [19], we can easily prove two properties of bu,v,w(30)bu,v,v=0forallu∈V,v∈H01Ω,(31)bu,v,w=−bu,w,vforallu∈V,v,w∈H01Ω.
Moreover, by the imbedding theoremH2Ω°C0Ω¯, we also can get that(32)∣bu,u,u0∣=∫Ωu·∇u·u0dx≤C∥u0∥H2Ω∥u∥H1Ω2,∣bB,B,B0∣=∫ΩB·∇B·B0dx≤C∥B0∥H2Ω∥B∥H1Ω2,∣bu0,u,u∣=∫Ωu0·∇u·udx≤C∥u0∥H2Ω∥u∥H1Ω2,∣bB0,B,B∣=∣∫ΩB0·∇B·Bdx∣≤C∥B0∥H2Ω∥B∥H1Ω2,with u0,B0∈H2Ω.
LetA= the space by ω1,ω2,⋯,ωn and the inner product ·,·A is inducted by that of V, namely, <·,·> given in (23). We choose X=A×A, and the inner product ·,·X is defined by(33)u,B,v,DX=u,vA+B,DA=<u,v>+<B,D>,∀u,B∈X,v,D∈X,where the norm ∥u,B∥X=∥u∥H1Ω2+∥B∥H1Ω21/2. We define P=Pn:X=A×A⟶X=A×A by(34)Pnu,B,v,DX=μ<u,v>+εbu0,u,v+εbu,u0,v+ε2bu,u,v−εbB0,B,v−εbB,B0,v−ε2bB,B,v−f1,v−λ<B,D>+εbB0,u,D+εbB,u0,D+ε2bB,u,D−εbu,B0,D−εbu0,B,D−ε2bu,B,D+f2,D,for any u,B,v,D∈X. The continuously of Pn is obvious. To verify (26) and (28), we can see with the help with (30), (31), and (32) that(35)Pnu,B,u,BX=μ<u,u>+εbu0,u,u+εbu,u0,u+ε2bu,u,u−εbB0,B,u−εbB,B0,u−ε2bB,B,u−f1,u−λ<B,B>−εbB0,u,B−εbB,u0,B−ε2bB,u,B+εbu,B0,B+εbu0,B,B+ε2bu,B,B+f2,B≥C∥u,B∥X2−∣εbu,u,u0∣−∣εbB0,B,u∣−∣εbB,B0,u∣−∣f1u∣−∣εbB0,u,B∣−∣εbB,u0,B∣−∣f2,B∣≥C∥u,B∥X2−εC′∥u0∥H2∥u∥H12−εC′∥B0∥H2∥B∥H1∥u∥H1−∥f1∥H−1∥u∥H1−εC′∥B0∥H2∥B∥H1∥u∥H1−εC′∥u0∥H2∥B∥H12−∥f2∥H−1∥B∥H1≥C∥u,B∥X2−εC′∥u0∥H2∥u,B∥X2−εC′∥B0∥H2∥u,B∥X2−∥f1∥H−1∥u,B∥X−∥f2∥H−1∥u,B∥X≥∥u,B∥XC−εC′∥u0∥H2−εC′∥B0∥H2∥u,B∥X−∥f1∥H−1−∥f2∥H−1,where C and C′ are positive numbers. Therefore, if we choose a small ε0, depending on ψ1,ψ2, such that(36)C−εC′∥u0∥H2−εC′∥B0∥H2>0,∀∣ε∣≤ε0,then Pnu,B,u,B>0 for ∥u,B∥X=k with(37)k>∥f1∥H−1+∥f2∥H−1C−εC′∥u0∥H2−εC′∥B0∥H2for∣ε∣≤ε0.
Hence, by Lemma (15), we can obtain the existence of un, Bn satisfying (26) and (28). Now, we would like to pass the limit of un, Bn. Multiplying (26) by ξ¯j,n and summing the corresponding equalities from 1 and n, we can get(38)μ<un,un>+εbu0,un,un+εbun,u0,un+ε2bun,un,un−εbB0,Bn,un−εbBn,B0,un−ε2bB0,Bn,un=f1,un.
Similarly, multiplying (28) by ζ¯j,n and summing the corresponding equalities from 1 and n gives(39)λ<Bn,Bn>−εbB0,un,Bn−εbBn,u0,Bn−ε2bBn,un,Bn(40)+εbun,B0,Bn+εbu0,Bn,Bn+ε2bun,Bn,Bn=f2,Bn.
Using (30)–(40), we can get that(41)∥un,Bn∥V≤C0∥f1∥H−1Ω+∥f2∥H−1Ω,where C0>0 is uniforming in ε provided ∣ε∣≤ε0. Therefore, there exists u,B in V and two subsequences un′, Bn′ such that(42)un′⇀uweaklyinVasn′⟶∞,(43)Bn′⇀BweaklyinVasn′⟶∞.
By the Sobolev imbedding theorem, we have that(44)un′⟶ustronglyinL4Ωasn′⟶∞,(45)Bn′⟶BstronglyinL4Ωasn′⟶∞.
By using (42)–(45), for any ω∈V, we can obtain that(46)∣bun′,un′,ω−bu,u,ω∣≤∣∫Ωun′·∇ω·un′−udx∣+∣∫Ωun′−u·∇ω·−udx∣≤∥un′∥L4∥∇ω∥L2∥un′−u∥L4+∥un′−u∥L4∥∇ω∥L2∥u∥L4⟶0asn′⟶∞.
Similarly, whenn′⟶∞ we can get that(47)bu0,un′,ω⟶bu0,u,ω,bun′,u0,ω⟶bu,u0,ω,bB0,Bn′,ω⟶bB0,B,ω,bBn′,B0,ω⟶bB,B0,ω,bBn′,Bn′,ω⟶bB,B,ω,bB0,un′,ω⟶bB0,u,ω,bBn′,u0,ω⟶bB,u0,ω,bBn′,un′,ω⟶bB,u,ω,bun′,B0,ω⟶bu,B0,ω,bu0,Bn′,ω⟶bu0,B,ω,bun′,Bn′,ω⟶bu,B,ω.with ω∈V.
Therefore, limitingun′, Bn′ in (26) and (28), we show that(48)μ<u,ωj>+εbu0,u,ωj+εbu,u0,ωj+ε2bu,u,ωj−εbB0,B,ωj−εbB,B0,ωj−ε2bB,B,ωj=f1,ωj,λ<B,ωj>−εbB0,u,ωj−εbB,u0,ωj−ε2bB,u,ωj+εbu,B0,ωj+εbu0,B,ωj+ε2bu,B,ωj=f2,ωj,for j=1,2,⋯. Since ω1,ω2,⋯ is a complete orthonormal system of V, we can obtain that(49)μ<u,ω>+εbu0,u,ω+εbu,u0,ω+ε2bu,u,ω−εbB0,B,ω−εbB,B0,ω−ε2bB,B,ω=f1,ω,λ<B,ω>−εbB0,u,ω−εbB,u0,ω−ε2bB,u,ω+εbu,B0,ω+εbu0,B,ω+ε2bu,B,ω=f2,ω,for all ω∈V. Thus, there exists p∈L2Ω (and u,B∈V) such that(50)−divσμu,p+εu0·∇u+εu·∇u0+ε2u·∇u−εB0·∇B−εB·∇B0−ε2B·∇B=f1,−divσ~λB−εB0·∇u−εB·∇u0−ε2B·∇u+εu·∇B0+εu0·∇B+ε2u·∇B=f2,in the weak sense.Now, the existence of weak solutionu,p,B to (19) is proved. To show the dependence of u,p,B on ε, we define u=u~ε, B=B~ε, and p=p~ε. Next, in order to show that the uniformly boundedness of u~ε,p~ε,B~ε about ε in some Sobolev space, we will discuss the regularity of u~ε,p~ε,B~ε. This uniformly boundedness of u~ε,p~ε,B~ε makes us to consider the limiting behavior of ε−1uε,ε−1pε,ε−1Bε. The proof of regularity for u~ε,p~ε,B~ε relies on the regularity result for the Stokes equations and the “bootstrapping” technique. Some arguments used here are derived from [19].Theorem 8.
Letu~ε,p~ε,B~ε be a weak solution of (19) for ∣ε∣≤ε0. Assume ε0<1, then u~ε,p~ε,B~ε∈H2Ω×H1Ω×H2Ω and satisfies(51)∥u~ε∥H2Ω+∥p~ε∥H1Ω/R+∥B~ε∥H2Ω≤Σj=14∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.Proof.
For conveniently, we denoteu=u~ε, p=p~ε, and B=B~ε. We now write the first equation and the second equation in the form of Stokes equation(52)divσμu,p=g1,divu=0,divσ~λB=g2,divB=0,with(53)g1=εu0·∇u+εu·∇u0+ε2u·∇u−εB0·∇B−εB·∇B0−ε2B·∇B−f1,g2=εB0·∇u+εB·∇u0+ε2B·∇u−εu·∇B0−εu0·∇B−ε2u·∇B−f2.
From the proof of the existence, we have thatu,B∈H01Ω and(54)∥u∥H1Ω≤C∥f1∥H−1Ω+∥f2∥H−1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2,∥B∥H1Ω≤C∥f1∥H−1Ω+∥f2∥H−1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2.
By the Sobolev imbedding, we can getu,B∈L6Ω. Subsequently, we obtain that u·∇u∈L3/2Ω, B·∇B∈L3/2Ω and(55)∥u·∇u∥L3/2Ω≤C∥u∥L6Ω∥∇u∥L2Ω≤C∥u∥H1Ω2≤C∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4,∥B·∇B∥L3/2Ω≤C∥B∥L6Ω∥∇B∥L2Ω≤C∥B∥H1Ω2≤C∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4.
Similarly, we can get(56)∥u0·∇u∥L3/2Ω≤C∥u0∥L6Ω∥∇u∥L2Ω≤C∥u0∥H1Ω∥u∥H1Ω≤C∥u0∥H2Ω∥u∥H1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4,∥B0·∇B∥L3/2Ω≤C∥B0∥L6Ω∥∇B∥L2Ω≤C∥B0∥H1Ω∥B∥H1Ω≤C∥B0∥H2Ω∥B∥H1Ω≤C∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4,∥u·∇u0∥L3/2Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4,∥B·∇B0∥L3/2Ω≤C∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4,∥f1∥L3/2Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2.
Therefore, asε0<1, we have that g1∈L3/2Ω and(57)∥g1∥L3/2Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4.
Similarly, from (22) and ε0<1, we have that g2∈L3/2Ω and(58)∥g2∥L3/2Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4.
The regularity theorem for the Stokes equations (Lemma5) implies that(59)∥u∥W2,3/2Ω+∥p∥W1,3/2Ω/R≤C∥g1∥L3/2Ω(60)≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4(61)∥B∥W2,3/2Ω≤C∥g2∥L3/2Ω(62)≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2+∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4.
The estimates (60) and (62) are not exactly what we want. We need to improve L3/2− base Sobolev norms to L2− one on the left-hand side of (60) and (62). This can be achieved by the “bootstrapping” argument. In view of Sobolev imbedding, W2,3/2Ω°LrΩ for any 1<r<∞, we thus obtain that u⊗u, B⊗B∈Ls for any s∈1,∞, and(63)∥u·∇u∥W−1,sΩ=∥divu⊗u∥W−1,sΩ≤C∥u⊗u∥LsΩ≤C∥u∥Lr1Ω∥u∥Lr2Ω(64)≤C∥u∥W2,3/2Ω∥u∥W2,3/2Ω(65)≤C′′∥ψ1∥H32∂Ω4+∥ψ2∥H32∂Ω4+∥ψ1∥H32∂Ω8+∥ψ2∥H32∂Ω8,where s−1=r1−1+r2−1 and C′′=C′′Ω,r1,r2. In the proof of (65), we uses the inequality (60) and divu=0. Similarly, by (62) and divB=0, we can get(66)∥B·∇B∥W−1,sΩ≤C∥B⊗B∥LsΩ≤C∥B∥Lr1Ω∥B∥Lr2Ω≤C∥B∥W2,3/2Ω2≤C′′∥ψ1∥H3/2∂Ω4+∥ψ2∥H3/2∂Ω4+∥ψ1∥H3/2∂Ω8+∥ψ2∥H3/2∂Ω8.
On the other hand,H2Ω°C0Ω¯, H2Ω=W2,2Ω°W2,3/2Ω. Likewise, we can get that(67)∥u0·∇u0∥W−1,sΩ≤C~∥ψ1∥H3/2∂Ω2+∑j=23∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥u·∇u0∥W−1,sΩ≤C~∥ψ1∥H3/2∂Ω2+∑j=23∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥B·∇B0∥W−1,sΩ≤C~∥ψ2∥H3/2∂Ω2+∑j=23∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥B0·∇B∥W−1,sΩ≤C~∥ψ2∥H3/2∂Ω2+∑j=23∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥f1∥W−1,sΩ≤C~∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2,
By (65)–(67), we can derive that(68)∥g1∥W−1,sΩ≤C∑j=13∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
Similarly, we can derive that(69)∥g2∥W−1,sΩ≤C∑j=13∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
Therefore, from Lemma4, we have that(70)∥u∥W1,sΩ+∥p∥LsΩ/R≤C∑j=13∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥B∥W1,sΩ≤C∑j=13∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,for any s∈1,∞.
Now we consider the estimates ofg1,g2 in L2Ω. From (70), we can get that u·∇u, B·∇B∈LsΩ for any s∈1,∞ and(71)∥u·∇u∥LsΩ≤C∥u∥Lr1Ω∥∇u∥Lr2Ω≤C∥u∥W1,r1Ω∥u∥W1,r2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,with s−1=r1−1+r2−1. Similarly, we can get(72)∥B·∇B∥LsΩ≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
In particular, we haveu·∇u, B·∇B∈L2Ω and(73)∥u·∇u∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,∥B·∇B∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
With Sobolev imbedding theorem, we can derive that(74)∥u0·∇u∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j+∥ψ1∥H3/2∂Ω2,∥u·∇u0∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j+∥ψ1∥H3/2∂Ω2,∥B0·∇B∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2,∥B·∇B0∥L2Ω≤C∑j=24∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2,∥f1∥L2Ω≤C∥f1∥H1Ω≤C∥ψ1∥H3/2∂Ω2+∥ψ2∥H3/2∂Ω2.
Then, from (73) and (74), we can get(75)∥g1∥L2Ω≤C∑j=14∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
Similarly, we have(76)∥g2∥L2Ω≤C∑j=14∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
Then, from Lemma5, we have(77)∥u∥H2Ω+∥p∥H1Ω/R≤C∑j=14∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j,(78)∥B∥H2Ω≤C∑j=14∥ψ1∥H3/2∂Ω2j+∥ψ2∥H3/2∂Ω2j.
From inequalities (77) and (78), we can get the inequality (51). The proof of Theorem 8 is completed.


## 3. Inverse Problem

In Section2,we have proved the existence of the solution u,p,B to the system (1) with the boundary data εψ1,εψ2. In this section, we derived the asymptotic behavior of solutions as ε⟶0 and prove the theorem in (1)–(5) by linearization technique.Given anyψ1,ψ2∈H3/2∂Ω, let u0,p0,B0∈H2Ω×H1Ω×H2Ω be the unique solution (p0 is unique up to a constant) of two Stokes equations(79)divσμu0,p0=0inΩ,divu0=0inΩ;(80)divσ~λB0=0inΩ,divB0=0inΩ.with boundary data u0,B0∂Ω=ψ1,ψ2. As proved in the previous section, there exists uε,pε,Bε with the form(81)uε=εu0+ε2u~ε,pε=εp0+ε2p~ε,Bε=εB0+ε2B~ε,satisfying (1) with boundary data uε,Bε∂Ω=εψ1,εψ2 for all ∣ε∣<ε0, where ε0 depends on ∥ψ1∥H3/2∂Ω and ∥ψ2∥H3/2∂Ω. Moreover, uε,pε,Bε statisfies (51). We immediately see that(82)∥ε−1uε−u0∥H2Ω=∥εu~ε∥H2Ω⟶0,∥ε−1pε−p0∥H1Ω/R=∥εp~ε∥H1Ω/R⟶0,∥ε−1Bε−B0∥H2Ω=∥εB~ε∥H2Ω⟶0as ε⟶0 and therefore(83)∥ε−1uε∂Ω−u0∂Ω∥H3/2∂Ω⟶0,(84)∥ε−1Bε∂Ω−B0∂Ω∥H3/2∂Ω⟶0,(85)∥ε−1σμuε,pεn∂Ω−σμu0,p0n∂Ω∥H1/2∂Ω⟶0,(86)∥ε−1σ~λBεn∂Ω−σ~λB0n∂Ω∥H1/2∂Ω⟶0,provided(87)∫Ωpεdx=∫Ωp0dx=0.As in [9], we define the Cauchy data associated to (79) and (80)(88)Cμ=u0∂Ω,σμu0,p0n∂Ω⊂H3/2∂Ω×H1/2∂Ω,Cλ=B0∂Ω,σ~λB0n∂Ω⊂H3/2∂Ω×H1/2∂Ω,where u0,p0 satisfies (79) and B0 satisfies (80). Now, the limits (83)–(86) lead to the following result.Theorem 9.
The Cauchy dataCμ, Cλ of Stokes equations can be uniquely determined from the Cauchy data C~μ,λ of the MHD equations.That is to say,Cμ1=Cμ2 and Cλ1=Cλ2 if C~μ1,λ1=C~μ2,λ2. So, we reduce the uniqueness question of the inverse problem for the MHD equations to that for the Stokes equations. Therefore, Theorems 1 and 2 follow from Theorem 9 and the unique determination of viscosity for the Stokes equation proved in [9].Theorem 10 (see [9], Theorem 1.1).
Assume thatμ1x and μ2x are two viscosity functions satisfying μ1,μ2∈Cn0Ω¯ for n0≥8 and(89)∂αμ1x=∂αμ2x∀x∈∂Ω,∣α∣≤1.
LetCμ1 and Cμ2 be the Cauchy data associated with μ1 and μ2, respectively. If Cμ1=Cμ2, then μ1=μ2.Theorem 11 (see [9], Corollary 1.4).
Let∂Ω be convex with nonvanishing Gauss curvature. Assume that μ1x and μ2x are two viscosity functions satisfying μ1,μ2∈Cn0Ω¯ for n0≥8. IfCμ1=Cμ2, then μ1=μ2.The regularity requirement in Theorem10 is to make sure that Eskin’s method [20] works in our case. We do not know whether it is optimal. Also, we define the Cauchy data of the Stokes equations as a subset of H1/2∂Ω×H−1/2∂Ω in [9], i.e., Cμ⊂H1/2∂Ω×H−1/2∂Ω. Nevertheless, the same proof in [9].In order to prove Theorem3, we cite the unique determination of viscosity for two-dimension Stokes equation which is proved in [12].Theorem 12 (see [12], Theorem 1.1).
LetΩ be a simply connected bounded domain in ℝ2 with smooth boundary. Suppose that μ1 and μ2 are two viscosity functions for Stokes equations, assume that μj∈C3Ω¯ and μj>0 with(90)∂αμ1∂Ω=∂αμ2∂Ωforallα≤1.
LetCμ1 and Cμ2 be the Cauchy data associated with μ1 and μ2, respectively. If Cμ1=Cμ2, then μ1=μ2 in Ω.Similar to Theorem9, we can prove that the conclusion of Theorem 9 is still true when u,B∈ℝ2 and Ω∈ℝ2. Then, we can use Theorem 9 and Theorem 12 to prove Theorem 3.



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