Local Well-posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. The free boundary problem for MHD is an important problem not only for mathematical fluid dynamics but also some application to the field of engineering In fact, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. In this paper, an electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in the maximal regularity class, and in particular, the regularity class of velocity fields is one more higher than the regularity class of magnetic fields. To prove our main result, we use Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions.


Introduction
In this paper, we prove the local well-posedness of a free boundary problem for the viscous nonhomogeneous incompressible magnetohydrodynamics. The problem here is formulated as follows: Let Ω + be a domain in the N -dimensional Euclidean space R N (N ≥ 2), and let Γ be the boundary of Ω + . Let Ω − be also a domain in R N whose boundary is Γ and S − . We assume that Ω + ∩ Ω − = ∅. Throughout the paper, we assume that Ω ± are uniform C 2 domains, that the weak Dirichlet problem is uniquely solvable in Ω + * and that dist (Γ, S − ) ≥ 2d − with some positive constants d − , where dist(A, B) denotes the distance of any two subsets A and B of R N defined by setting dist(A, B) = inf{|x−y| | x ∈ A, y ∈ B}.
Let Ω = Ω + ∪ Γ ∪ Ω − andΩ = Ω + ∪ Ω − . The boundary of Ω is S − . We may consider the case that S − is an empty set, and in this case Ω = R N . Physically, we consider the case where Ω + is filled by a non-homogeneous incompressible magnetohydrodynamic (MHD) fluid and Ω − is filled by an insulating gas. We consider a motion of an MHD fluid in a time dependent domain Ω t+ whose boundary is Γ t subject to an electomagnetic field generated in a domain Ω t− = Ω \ (Ω t+ ∪ Γ t ) by some currents located on a fixed boundary S − of Ω t− . Let n t be the unit outer normal to Γ t oriented from Ω t+ into Ω t− , and let n − be respective the unit outer normals to S − . Given any functions, v ± , defined on Ω t± , v is defined by v(x) = v ± (x) for x ∈ Ω t± for t ≥ 0, where Ω 0± = Ω ± . Moreover, what v = v ± denotes that v(x) = v + (x) for x ∈ Ω t+ and v(x) = v t− (x) for x ∈ Ω t− . Let [[v]](x 0 ) = lim for every point x 0 ∈ Γ t , which is the jump quantity of v across Γ.
The purpose of this paper is to prove the local well-posedness of the free boundary problem formulated by the set of the following equations: (1.1) Here, x, t), . . . , v N (x, t)) ⊤ is the velocity vector field, where M ⊤ stands for the transposed M , p = p(x, t) the pressure fields, and H = H ± = (H ±1 (x, t), . . . , H ±N (x, t)) ⊤ the magnetic vector field. The v, p, and H are unknows, while v 0 and H 0 are prescribed N -component vectors of functions. As for the remaining symbols, T(v, p) = νD(v) − pI is the viscous stress tensor, D(v) = ∇v + (∇v) ⊤ is the doubled deformation tensor whose (i, j)th component is ∂ j v i + ∂ i v j with ∂ i = ∂/∂x i , I the N × N unit matrix, T M (H + ) = µ + (H + ⊗ H + − 1 2 |H + | 2 I) the magnetic stress tensor, curl v = (∇v) ⊤ − ∇v the doubled rotation tensor whose (i, j)th component is ∂ j v i − ∂ i v j , V Γt the velocity of the evolution of Γ t in the direction of n t . Moreover, ρ, µ ± , ν, and α ± are positive constants describing respective the mass density, the magnetic permability, the kinematic viscosity, and the conductivity. Finally, for any matrix field K with (i, j)th component K ij , the quantity Div K is an N -vector of functions with the ith component N j=1 ∂ j K ij , and for any N -vectors of functions u = (u 1 , . . . , u N ) ⊤ and w = (w 1 , . . . , w N ) ⊤ , div u = N j=1 ∂ j u j , u · ∇w is an N -vector of functions with the ith component N j=1 u j ∂ j w i , and u ⊗ w an N × N matrix with the (i, j)th component u i w j . We notice that in the three dimensional case ∆v = −Div curl v + ∇div v, Div (v ⊗ H − H ⊗ v) = vdiv H − Hdiv v + H · ∇v − v · ∇H, rot rot H = Div curl H, rot (v × H) = Div (v ⊗ H − H ⊗ v), (1.2) where × is the exterior product. In particular, in the three dimensional case, the set of equations for the magnetic vector field in equations (1. This is a standard description, and so the set of equations for the magnetic field in equations (1.1) is the N -dimensional mathematical description of equations for the magnetic vector field with transmission conditions.
In equations (1.1), there is one equation for the magnetic fields H ± too many, so that in this paper instead of (1.1), we consider the following equations: Namely, two equations: div H ± = 0 in Ω T ± is replaced with one transmission condition: [[µdiv H]] = 0 on Γ T . Employing the same argument as in Frolova and Shibata [7,Appendix], we see that in equations (1.3) if div H = 0 initially, then div H = 0 inΩ follows automatically for any t > 0 as long as solutions exist. Thus, the local well-posedness of equations (1.1) follows from that of equations (1.3) provided that the initial data H 0± satisfy the divergence zero condition: div H 0± = 0. This paper devotes to proving the local well-posedness of equations (1.3) in the maximal L p -L q regularity framework.
The MHD equations can be found in [1,10]. The solvability of MHD equations was first obtained by Ladyzhenskaya and Solonnikov [11]. The initial-boundary value problem for MHD equations with non-slip conditions for the velocity vector field and perfect wall conditions for the magnetic vector field was studied by Sermange and Temam [15] in a bounded domain and by Yamaguchi [26] in an exterior domain. In their studies [15,26], the boundary is fixed. On the other hand, in the field of engineering, when a thermonuclear reaction is caused artificially, a high-temperature plasma is sometimes subjected to a magnetic field and held in the air, and the boundary of the fluid at this time is a free one. From this point of view, free boundary problem for MHD equations is important. The local well-posedness for free boundary problem for MHD equations was first proved by Padula and Solonnikov [13] in the case where Ω t+ is a bounded domain surrounded by a vacuum area, Ω t− . In [13], the solution was obtained in Sobolev-Slobodetskii spaces in the L 2 framework of fractional order greater than 2. Later on, the global well-posedness was proved by Frolova [5] and Solonnikov and Frolova [25]. Moreover, the L p approach to the same problem was done by Solonnikov [22,23]. When Ω t+ is a bounded domain which is surrounded by an electromagnetic field generated in a domain, Ω t− , Kacprzyk proved the local well-posedness in [8] and global well-posedness in [9]. In [8,9], the solution was also obtained in Sobolev-Slobodetskii spaces in the L 2 framework of fractional order greater than 2.
Recently, the L p -L q maximal regularity theorem for the initial boundary value problem of the system of parabolic equations with non-homogeneous boundary conditions have been studied by using R-solver in [19] and references therein and by using H ∞ calculus in [14] and references therein. They are completely different approaches. In particular, Shibata [16,18] proved the L p -L q maximal regularity for the Stokes equations with non-homogeneous free boundary conditions by R-solver theory and Frolova and Shibata [7] proved it for linearized equations for the magnetic vecor fields with transmission conditions on the interface and perfect wall conditions on the fixed boundary arising in the study of two phase problems for the MHD flows also by using R-solver. The results in [16,18,7] enable us to prove the local well-posedness for equations (1.3) in the L p -L q maximal regularity class.
Aside from dynamical boundary conditions on Γ t , a kinematic condition, V Γt = v · n t , is satisfied on Γ t , which represents Γ t as a set of points x = x(ξ, t) for ξ ∈ Γ, where x(ξ, t) is the solution of the Cauchy problem: This expresses the fact that the free surface Γ t consists for all t > 0 of the same fluid particules, which do not leave it and are not incident on it from inside Ω t+ . Problem (1.3) can be written as an initial-boundary problem with transmission conditions on Γ if we go over the Euler coordinates x ∈Ω t = Ω t+ ∪ Ω t− to the Lagrange coordinates ξ ∈Ω = Ω + ∪ Ω − connected with x by (1.4). Since the velocity field, u + (ξ, t) = v(x, t), is given only in Ω + , we extend it to u − defined on Ω − in such a way that (1.5) Let ϕ(ξ) be a C ∞ (R N ) function which equals 1 when dist (ξ, S − ) ≥ 2d − and equals 0 when dist (ξ, S − ) ≤ d − . The connection between Euler coordinates x and Lagrangian coordinates ξ is defined by setting is transformed by (1.6) to the following equations: Here, n is the unit outer normal to Γ oriented from Ω + into Ω − , d τ = d− < d, n > n for any N -vector d, and the N 1 (u + ,H + ), . . . , N 9 (u,H) are nonlinear terms defined in Sect. 2 below. Our main result is the following theorem.
Theorem 1. Let 1 < p, q < ∞ and B ≥ 1. Assume that 2/p + N/q < 1, that Ω + is a uniform C 3 domain and Ω = Ω + ∪ Γ ∪ Ω − a uniform C 2 domain, and that the weak problem is uniquely solvable in Ω + for q and q ′ = q/(q − 1). Let initial data u 0+ and H 0± with satisfy the conditions: and compatibility conditions div u 0+ = 0 in Ω + , Then, the there exists a time T > 0 for which problem (1.7) admits unique solutions u + andH ± with possessing the estimate: with some polynomial f (B) with respect to B.

Remark 2.
As was mentioned after equations (1.3), if we assume that divH 0± = 0 in Ω ± in addition, then div H ± = 0 in Ω T ± , and so v and H are solutions of equations (1.1). Thus, we obtain the local well-posedness of equations (1.1) from Theorem 1.
For 1 ≤ q ≤ ∞, m ∈ N, s ∈ R, and any domain D ⊂ R N , we denote the standard Lebesgue space, Sobolev space, and Besov space by L q (D), H m q (D), and B s q,p (D) respectively, while · Lq(D) , · H m q (D) , and · B s q,p (D) denote their norms. We write W s q (D) = B s q,q (D) and H 0 q , B s q,p }, the function spaces H(Ω) (Ω = Ω + ∪ Ω − ) and their norms are defined by setting For any Banach space X, · X being its norm, X d denotes the d product space defined by {x = (x 1 , . . . , x d ) | x i ∈ X}, while the norm of X d is simply written by · X , which is defined by setting x X = d j=1 x j X . For any time interval (a, b), L p ((a, b), X) and H m p ((a, b), X) denote respective the standard X-valued Lebesgue space and X-valued Sobolev space, while · Lp((a,b),X) and · H m p ((a,b),X) denote their norms. Let F and F −1 be respective the Fourier transform and the Fourier inverse transform. Let H s p (R, X), s > 0, be the Bessel potential space of order s defined by Let a · b =< a, b >= N j=1 a j b j for any N -vectors a = (a 1 , . . . , a N ) and b = (b 1 , . . . , b N ). For any N -vector a, let a τ := a− < a, n > n. For any two N × N -matrices A = (A ij ) and B = (B ij ), the quantity A : B is defined by A : For any domain G with boundary ∂G, we set where v(x) is the complex conjugate of v(x) and dσ denotes the surface element of ∂G. Given 1 < q < ∞, let q ′ = q/(q − 1). Throughout the paper, the letter C denotes generic constants and C a,b,··· the constant which depends on a, b, · · · . The values of constants C, C a,b,··· may be changed from line to line. When we describe nonlinear terms N 1 (u + ,H + ), . . . , N 9 (u,H) in (1.7), we use the following notational conventions. Let u i (i = 1, . . . , m) be n i -vectors whose jth component is u ij , and then u 1 ⊗ · · · ⊗ u m denotes an n = m i=1 n i vector whose (j 1 , . . . , j m )th component is Π m i=1 u ij i and the set {(j 1 , . . . , j m ) | 1 ≤ j i ≤ n i , i = 1, . . . , m} is rearranged as {k | k = 1, 2, . . . , n} and k is the corresponding number to some (j 1 , . . . , j m ). For example, u ⊗ ∇v is an N + N 2 vector whose (i, j, k) component is . . , m ℓ , ℓ = 1, . . . , n) be n ℓ i -vectors, let A ℓ be n ℓ × N matrices, where n ℓ = m ℓ i=1 n ℓ i , and set A = {A 1 , . . . , A m }. And then, we write When there are two sets of matrices A = {A 1 , . . . , A n } and B = {B 1 , . . . , B n }, we write
In partuclar, we have the fourth equation in (1.7). We now consider the transmission conditions. The unit outer normal, n t , to the Γ t is represented by Choosing δ > 0 small enough, we may write Choosing δ > 0 small if necessary, we may assume that (I + V n (k)) −1 exists and we may write Using notational convention defined in Notation, we may write where A 4 (k) is a set of matrices of functions consisting of products of elements of n and smooth functions defined for |k| ≤ δ. By (2.11) and (2.18), Thus, setting Using notational convention defined in Notation and noting that [[Ψ u ]] = 0 on Γ as follows from (1.5), we may write where A 61 (k) and A 62 (k) are a matrix and a set of matrices of functions consisting of products of elements of n and smooth functions defined for |k| ≤ δ, and B is a set of matrices of functions such that Bu + ⊗H + = µ + (u + ⊗H + −H + ⊗ u + )n. In particular, and so setting where A 7 (k) is a matrix of functions consisting of products of elements of n and smooth functions defined for |k| ≤ δ. Notice that and so , setting and so, setting For notational simplicity, we set where A 8 (k) is a set of matrices of functions consisting of products of elements of n and smooth functions defined for |k| ≤ δ. Notice that

The Stokes equations with free boundary conditions
This subsection is devoted to presenting the L p -L q maximal regularity theorem for the Stokes equations with free boundary conditions. The problem considered here is formulated by the following equations: To state assumptions for equations (3.1), we make two definitions.

Definition 3.
Let Ω + be a domain given in Introduction. We say that Ω + is a uniform C 3 domain, if there exist positive constants a 1 , a 2 , and A such that the following assertion holds: For any Here, we have set LetĤ 1 q,0 (Ω + ) be a homogeneous Sobolev space defined by lettinĝ is called the weak Dirichlet problem, where q ′ = q/(q − 1).

Definition 4.
We say that the weak Dirichlet problem (3.3) is uniquely solvable for an index q if for any f ∈ L q (Ω) N , problem (3.3) admits a unique solution u ∈Ĥ 1 q,0 (Ω) possessing the estimate: Let J q (Ω + ) be the set of all solenoidal vector of functions.
(Ω) and let f , g, g, h be functions appearing in equations (3.1) satisfying the following conditions: Assume that u 0 , g, and h satisfy the following compatibility conditions: for some positive constants C and γ 1 independent of T .

Remark 6.
(1) Theorem 5 has been proved by Shibata [16] in the standard case where But, in Theorem 5 one more additional regularity is stated, which is necessary for our approach to prove the well-posedness of equations (1.3). The idea of proving how to obtain third order regularity of the fluid vector field will be given in Appendix below.
(2) The uniqueness holds in the following sense.
satisfy the homogeneous equations: then v = 0 and q = 0.

Two phase problem for the linear electro-magnetic vector field equations
This subsection is devoted to presenting the L p -L q maximal regularity due to Frolova and Shibata [7] for the linear electro-magnetic vector field equations. The problem is formulated by a set of the following equations: To state the main result, we make a definition.

Definition 7.
Let Ω = Ω + ∪ Γ ∪ Ω − be a domain given in Introduction. We say that Ω is a uniform C 2 domain with interface Γ if there exist positive constants a 1 , a 2 , and A such that the following assertion holds: For any and for any Here, we have set Theorem 8. Let 1 < p, q < ∞, 2/p + N/q = 1, 2, and T > 0. Assume that Ω is a uniform C 2 domain with interface Γ. Then, there exists a γ 2 such that the following assertion holds: and k N are functions given in the right side of (3.6), and the following conditions hold: for any γ ≥ γ 2 . Moreover, we assume that H 0 , h and k satisfy the following compatibility conditions: provided 2/p + N/q < 1; possessing the estimate: Remark 9. (1) Theorem 8 was proved by Froloba and Shibata [6].
(2) The uniqueness holds in the following sense. Let H with satisfies the homogeneous equations: then H = 0 inΩ × (0, T ).

Estimate of non-linear terms
Let u + and H ± be N -vectors of functions such that and we shall estimate nonlinear terms N 1 (u + , H + ), . . . , N 9 (u, H) appearing in the right side of equations (1.7). Here, w = w + for x ∈ Ω + and w = w − for x ∈ Ω − (w ∈ {u, H}) and u − is an extension of u + defined in (1.5). For notational simplicity, we set Moreover, let u i + and H i ± (i = 1, 2) be N -vectors of functions such that We also consider the differences: . Here, w = w + for x ∈ Ω + and w = w − for x ∈ Ω − (w ∈ {u 1 , u 2 , H 1 , H 2 }) and u i − is an extension of u i + defined in (1.5). For notational simplicity, we assume that for some constant B > 0. In what follows, we assume that 2 < p < ∞, N < q < ∞ and 2/p + N/q < 1.
To estimate nonlinear terms, we use the following inequalities which follows from Sobolev's inequality.

A proof of Theorem 1
We shall prove Theorem 1 by contraction mapping principle. For this purpose, we define an underlying space U T,L for a large number L > 1 and a small time T ∈ (0, 1) by setting Next, let N 1 (u + , H + ), N 2 (u + ), N 3 (u + ) and N 4 (u + , H + ) be respective non-linear terms given in (2.15), in Ω + .
(A.2) † Here, we just give an idea of obtaining third order regularities. An idea also is found in [2, Appendix 6.2]. To prove Theorem 5 exactly from the R-bounded solution operators point of view, we have to start returning the non-zero f , g and g situation to the situation where f = g = g = 0, which needs an idea. We will give an exact proof of Theorem 5 in a forthcomming paper.
We now prove that u ∈ H 3 q (R N + ) N and ∇p ∈ H 1 q (R N + ) N provided that f ∈ H 1 q (R N + ) N , g ∈ H 2 q (R N + ), g ∈ H 1 q (R N + ) N , and h ∈ H 2 q (R N + ) N . Moreover, u and p satisfy the estimate: In fact, differentiating equations (A.1) with respect to tangential variables x j (j = 1, . . . , N − 1) and noting that ∂ j u and ∂ j p satisfy equations replacing f , g = div g, and h with ∂ j f , ∂ j g = div ∂ j g and ∂ j h, by (A.2) and the uniquness of solutions we see that ∂ j u ∈ H 2 q (R N + ) N and ∇∂ j p ∈ L q (R N + ) N , and for j = 1, . . . , N − 1. To estimate ∂ N u and ∂ N p, we start with estimating ∂ N u N . In fact, from the divergence equations it follows that ∂ N u N = − N −1 j=1 ∂ j u j + g, and so which, combined with (A.4) yields that ( ∂ j f Lq(R N + ) + (∂ j g, ∂ j h) H 1 q (R N + ) + λ 1/2 (∂ j g, ∂ j h) Lq(R N + ) + λ∂ j g Lq (R N + ) ) + (λg, ∂ 2 N g) Lq(R N + ) .

(A.5)
From the the N -th component of the first equation of equations (A.1) and div u = g, we have and so, we see that ∂ 2 N p ∈ L q (R N + ) and From equations (A.1), we have Differentiating the first equation of the above set of equations with respect to x N and setting ∂ N u j = v, we have λv − µ∆v = ∂ N f j − ∂ j ∂ N p + µ∂ N ∂ j g in R N + , v = −∂ j u N + µ −1 h j on R N 0 . Thus, setting w = v + ∂ j u N − µ −1 h j , we have Thus, by a known estimate for the Dirichlet problem, we have (A.7) Noting that λ 1/2 (∂ j g, ∂ j h) Lq(R N + ) ≤ C( λ(g, h) Lq (R N + ) + (g, h) H 2 q (R N + ) ) and combining (A.2) and (A.4)-(A.7), we have (A.3).
Localizing the problem and using the argument above, we can show Theorem 5.