Inverse Boundary Value Problem for the Magnetohydrodynamics Equations

Abstract

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

1. Introduction

Let be an open bounded domain with boundary . Assume that is filled with an incompressible fluid. and stand for velocity field and magnetic field, respectively. satisfies the following stationary magnetohydrodynamics(MHD) equations:whereand is the symmetric part of the matrix . is the pressure. The notation is the identity matrix. Here, is the viscosity function and 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) bywhere is the unit outer normal of the boundary and , and satisfies the compatibility conditions

In the physical sense, stands for the stress acting on and is called the Cauchy force. The motivation of this paper is to determine from the knowledge of the Cauchy data .

To discuss the inverse problem, we will not consider the general Dirichlet data . Particularly, we shall assumewith sufficiently small and satisfying the compatibility condition (4). For such a choice of Dirichlet data, we can obtain that there exists a solution of the equation (1) with , and the boundary trace . Thus, the Cauchy data is meaningful in this case. When is sufficiently small, we even know that the solution to (1) is unique ( 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. Let . Assume that and are viscosity functions and and are magnetic diffusivity functions. for andLet and be the Cauchy data with and , respectively. If , then .
When the boundary is convex and has nonvanishing Gauss curvature, we can remove the assumption (6)(7) from Theorem 1.

Theorem 2. Let . Suppose that is convex with the nonvanishing Gauss curvature, and and are viscosity functions and magnetic diffusivity functions, respectively, satisfying for . If , then .

Theorem 3. Let be a simply connected bounded domain in with smooth boundary. Suppose that and are viscosity functions and magnetic diffusivity functions, respectively. Assume that and withLet and be the Cauchy data associated with and , respectively. If , then .

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 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 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 as in (5). The key step in the proofs of Theorems 1, 2, and 3 is to prove the existence of the solution to (1) with boundary condition (5) and converging to in suitable Sobolev spaces, where and satisfy two Stokes equations, respectively, whereafter the Cauchy data of MHD equation can determine Cauchy data of the Stokes equation about and of the Stokes equation about . 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 Section 2, we will prove the existence of the boundary value problem for (1). In Section 3, we linearize the Cauchy data 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:with and the compatibility conditions (4). When and 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). Suppose . For any,there exists a unique solution ( is unique up to a constant) satisfyingMoreover, this solution obeys the estimatewhere depends on and .

Lemma 5 (see [14], Theorem 11). Suppose with the integer , . For any,there exists a unique solution ( is unique up to a constant) satisfying (11). Moreover, this solution obeys the estimatewhere depends on , min, and .

Lemma 6 (see [19], Lemma 1.4, Chapter II). Let be a finite-dimensional Hilbert space with inner product and norm . And let be a continuous map from to itself such thatThen, there exists with so that .
As presented in the introduction, we assume with and look for satisfies (9). Then, the problem (9) is reduced toWe will prove the problem (16) with the solution , , , where satisfies the Stokes equation satisfiesBy Lemma 5, we can know that for each there exists a unique ( is unique up to a constant) satisfying (17) and (18) and the estimatewith . In view of the Sobolev imbedding theorems and , we get thatIn order to solve the problem (19), we first prove the existence of (19).

Theorem 7. There exists a positive number depending on such that for any , problem (19) has at least one weak solution .

Proof. Similar to Chapter II, Section 1 in [19], we use the Galerkin method to solve the problem (19). We denote , . . By Korn’s inequality and Poincare’s inequality, it is easily to prove that is a separable Hilbert space with respect to the inner productNote that is a closed subspace of , which is also separable. Let be elements of which form a complete orthonormal system of , whereLet , with satisfyfor , whereAnalogous to Lemma 1.3, Chapter II in [19], we can easily prove two properties of Moreover, by the imbedding theorem , we also can get thatwith .
Let the space by and the inner product is inducted by that of , namely, given in (23). We choose , and the inner product is defined bywhere the norm . We define byfor any . The continuously of is obvious. To verify (26) and (28), we can see with the help with (30), (31), and (32) thatwhere and are positive numbers. Therefore, if we choose a small , depending on ,, such thatthen for withHence, by Lemma (15), we can obtain the existence of , satisfying (26) and (28). Now, we would like to pass the limit of , . Multiplying (26) by and summing the corresponding equalities from 1 and , we can getSimilarly, multiplying (28) by and summing the corresponding equalities from 1 and givesUsing (30)–(40), we can get thatwhere is uniforming in provided . Therefore, there exists in and two subsequences , such thatBy the Sobolev imbedding theorem, we have thatBy using (42)–(45), for any , we can obtain thatSimilarly, when we can get thatwith .
Therefore, limiting , in (26) and (28), we show thatfor Since is a complete orthonormal system of , we can obtain thatfor all . Thus, there exists (and ) such thatin the weak sense.

Now, the existence of weak solution to (19) is proved. To show the dependence of on , we define , , and . Next, in order to show that the uniformly boundedness of about in some Sobolev space, we will discuss the regularity of . This uniformly boundedness of makes us to consider the limiting behavior of . The proof of regularity for relies on the regularity result for the Stokes equations and the “bootstrapping” technique. Some arguments used here are derived from [19].

Theorem 8. Let be a weak solution of (19) for . Assume , then and satisfies

Proof. For conveniently, we denote , , and . We now write the first equation and the second equation in the form of Stokes equationwithFrom the proof of the existence, we have that andBy the Sobolev imbedding, we can get . Subsequently, we obtain that , andSimilarly, we can getTherefore, as , we have that andSimilarly, from (22) and , we have that andThe regularity theorem for the Stokes equations (Lemma 5) implies thatThe estimates (60) and (62) are not exactly what we want. We need to improve base Sobolev norms to one on the left-hand side of (60) and (62). This can be achieved by the “bootstrapping” argument. In view of Sobolev imbedding, for any , we thus obtain that , for any , andwhere and . In the proof of (65), we uses the inequality (60) and . Similarly, by (62) and , we can getOn the other hand, , . Likewise, we can get thatBy (65)–(67), we can derive thatSimilarly, we can derive thatTherefore, from Lemma 4, we have thatfor any .
Now we consider the estimates of in . From (70), we can get that , for any andwith . Similarly, we can getIn particular, we have , andWith Sobolev imbedding theorem, we can derive thatThen, from (73) and (74), we can getSimilarly, we haveThen, from Lemma 5, we haveFrom inequalities (77) and (78), we can get the inequality (51). The proof of Theorem 8 is completed.

3. Inverse Problem

In Section 2,we have proved the existence of the solution to the system (1) with the boundary data . In this section, we derived the asymptotic behavior of solutions as and prove the theorem in (1)–(5) by linearization technique.

Given any , let be the unique solution ( is unique up to a constant) of two Stokes equationswith boundary data . As proved in the previous section, there exists with the formsatisfying (1) with boundary data for all , where depends on and . Moreover, statisfies (51). We immediately see thatas and thereforeprovided

As in [9], we define the Cauchy data associated to (79) and (80)where satisfies (79) and satisfies (80). Now, the limits (83)–(86) lead to the following result.

Theorem 9. The Cauchy data , of Stokes equations can be uniquely determined from the Cauchy data of the MHD equations.

That is to say, and if . 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 that and are two viscosity functions satisfying for andLet and be the Cauchy data associated with and , respectively. If , then .

Theorem 11 (see [9], Corollary 1.4). Let be convex with nonvanishing Gauss curvature. Assume that and are two viscosity functions satisfying for . If, then .

The regularity requirement in Theorem 10 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 in [9], i.e., . Nevertheless, the same proof in [9].

In order to prove Theorem 3, we cite the unique determination of viscosity for two-dimension Stokes equation which is proved in [12].

Theorem 12 (see [12], Theorem 1.1). Let be a simply connected bounded domain in with smooth boundary. Suppose that and are two viscosity functions for Stokes equations, assume that and withLet and be the Cauchy data associated with and , respectively. If , then in .

Similar to Theorem 9, we can prove that the conclusion of Theorem 9 is still true when and . Then, we can use Theorem 9 and Theorem 12 to prove Theorem 3.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work of L. H He and Z. Tan was supported by the National Natural Science Foundation of China (11531010, 11726023).