Inverse Boundary Value Problem for the Magnetohydrodynamics Equations

In this paper, we consider the stationary magnetohydrodynamics (MHD) equations in a bounded domain of 
 
 
 
 ℝ
 
 
 d
 
 
  
 
 
 d
 =
 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.


Introduction
Let Ω ⊂ ℝ d ðd = 2, 3Þ be an open bounded domain with boundary ∂Ω ∈ C ∞ . Assume that Ω is filled with an incompressible fluid. u ∈ ℝ d and B ∈ ℝ d stand for velocity field and magnetic field, respectively. ðu, BÞ satisfies the following stationary magnetohydrodynamics(MHD) equations: where σ μ u, p ð Þ= 2μ Sym ∇u ð Þ− pI ;σ λ B ð Þ = 2λ Sym ∇B ð Þ, ð2Þ and SymðAÞ = ðA + A T Þ/2 is the symmetric part of the matrix A. p is the pressure. The notation I is the identity matrix.
Here, μðxÞ > 0 is the viscosity function and λðxÞ > 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) bỹ where n is the unit outer normal of the boundary ∂Ω and uj ∂Ω , and Bj ∂Ω satisfies the compatibility conditions In the physical sense, σ μ ðu, pÞnj ∂Ω stands for the stress acting on ∂Ω and is called the Cauchy force. The motivation of this paper is to determine ðμ, λÞ from the knowledge of the Cauchy dataC μ,λ .
To discuss the inverse problem, we will not consider the general Dirichlet data ðu, BÞj ∂Ω = ðϕ 1 , ϕ 2 Þ. Particularly, we shall assume with |ε | sufficiently small and ψ i ∈ H 3/2 ð∂ΩÞði = 1, 2Þ satisfying the compatibility condition (4). For such a choice of Dirichlet data, we can obtain that there exists a solution ðu, p, BÞ of the equation (1) with uj ∂Ω = εψ 1 , Bj ∂Ω = εψ 2 and the boundary trace σ μ ðu, pÞnj ∂Ω ,σ λ ðBÞnj ∂Ω ∈ H 1/2 ð∂ΩÞ. Thus, the Cauchy dataC μ,λ is meaningful in this case. When |ε | is sufficiently small, we even know that the solution ðu, p, BÞ 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.
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][3][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][7][8]), complex exponential solution (see [9][10][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][15][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 ðu ε , p ε , B ε Þ to (1) with boundary condition (5) and  (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 Section 2, we will prove the existence of the boundary value problem for (1). In Section 3, we linearize the Cauchy dataC μ,λ and prove Theorems 1, 2, and 3.

Direct Problem
In this section, we would like to prove the existence of the boundary value problem: with ϕ i ði = 1, 2Þ ∈ H 3/2 ð∂ΩÞ and the compatibility conditions (4). When μ and λ are constants, this problem has been discussed in literature [18].

Journal of Function Spaces
In order to prove the existence of equation (9), we first introduce some lemmas.
Lemma 6 (see [19], Lemma 1.4, Chapter II). Let X be a finite-dimensional Hilbert space with inner product ½·, · and norm ∥·∥. And let P be a continuous map from X to itself such that Then, there exists ζ ∈ X with ∥ζ∥≤k so that PðζÞ = 0.
Proof. Similar to Chapter II, Section 1 in [19], we use the Galerkin method to solve the problem (19). We denote V = fu ∈ H 1 0 ðΩÞ ; div u = 0g, V = V × V. ∥ðu, BÞ∥ V = ð∥u∥ 2 H 1 ðΩÞ +∥B∥ 2 H 1 ðΩÞ Þ 1/2 . By Korn's inequality and Poincare's inequality, it is easily to prove that H 1 0 ðΩÞ is a separable Hilbert space with respect to the inner product Note that V is a closed subspace of H 1 0 ðΩÞ, which is also separable. Let ω 1 , ω 2 , ⋯ be elements of V which form a com- Let u n = ∑ n j=1 ξ j,n ω j , B n = ∑ n j=1 ζ j,n ω j with ξ j,n , ζ j,n ∈ ℂ satisfy for ð29Þ Analogous to Lemma 1.3, Chapter II in [19], we can easily prove two properties of bðu, v, wÞ Moreover, by the imbedding theorem H 2 ðΩÞ°C 0 ð ΩÞ, we also can get that with u 0 , B 0 ∈ H 2 ðΩÞ. Let A = the space by ω 1 , ω 2 , ⋯, ω n and the inner product ½·, · A is inducted by that of V, namely, <·, · > given in (23). We choose X = A × A, and the inner product ½·, · X is defined by where the norm ∥ðu, BÞ∥ X = ð∥u∥ 2 H 1 ðΩÞ +∥B∥ 2 where C and C ′ are positive numbers. Therefore, if we choose a small ε 0 , depending on ψ 1 ,ψ 2 , such that then ½P n ðu, BÞ, ðu, BÞ > 0 for ∥ðu, BÞ∥ X = k with Hence, by Lemma (15), we can obtain the existence of u n , B n satisfying (26) and (28). Now, we would like to pass the limit of u n , B n . Multiplying (26) by ξ j,n and summing the corresponding equalities from 1 and n, we can get Similarly, multiplying (28) by ζ j,n and summing the corresponding equalities from 1 and n gives Using (30)-(40), we can get that where C 0 > 0 is uniforming in ε provided |ε | ≤ε 0 . Therefore, there exists ðu, BÞ in V and two subsequences fu n ′ g, fB n ′ g such that By the Sobolev imbedding theorem, we have that By using (42)-(45), for any ω ∈ V , we can obtain that ⟶0 as n′⟶∞ :