On a free boundary problem of magnetohydrodynamics in multi-connected domains

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1 On a free boundary problem of magnetohydrodynamics in multi-connected domains V.A.Solonnikov June 212, Frauenhimsee 1 Formulation of the problem Domains 1t : variable domain filled with a liquid, 2t : vacuum region surrounding 1t, 3 : fixed domain where the electric current j(x, t) is given; = 1t 3 2t, Γ t = 1t : free surface, S 3 : the boundary of 3, S: the boundary of a perfect conductor. Equations of magnetohydrodynamics. Navier-Stokes equations: { vt + (v )v (T (v, p) + T M (H)) = f(x, t), v(x, t) =, x 1t, t >, Maxwell equations: µh t = rote, H =, x 1t 2t 3, roth = α(e + µ(v H)), x 1t, roth =, H =, E =, x 2t, roth = αe + j(x, t), x 3. (1.1) (1.2) v, p: velocity and pressure, f: mass forces, H(x, t), E(x, t): magnetic and electric fields, µ: piecewise constant positive function equal to µ i in i α: piecewise constant function positive in 1t, 3 and α = in 2t, T (v, p) = pi + νs(v): the viscous stress tensor, S(v) = v + ( v) T : the doubled rate-of-strain tensor, T M (H) = µ(h H 1 2 H 2 I): the magnetic stress tensor. 1

2 Boundary and jump conditions: H n =, E τ =, x S, [µh n] =, [H τ ] =, [E τ ] =, x S 3, ( T (v, p) + [TM (H)] ) n =, V n = v n, [µh n] =, [H τ ] =, n t [µ]h τ + [n x E] =, x Γ t, (1.3) H τ = H n(n H), E τ = E n(n E): tangential components of H and E, [F ]: jump of the vector field F (x) on Γ t and S 3, V n : the velocity of evolution of Γ t in the direction of the exterior normal n, n = (n x, n t ): normal to the surface x Γ t, t (, T ) in R 4. Initial and orthogonality conditions v(x, ) = v (x), x 1, H(x, ) = H (x), x 1 2 3, (1.4) S k E nds =. (1.5) S k : connected components of the boundary of 2t, i = i, i = 1, 2, 3 Compatibility conditions v (x) =, x 1, H (x) =, x 1 2 3, roth (x) =, x 2, [(S(v )n) τ ] =, [H τ ] =, [µh N] =, x Γ, [H τ ] =, [µh n] =, x S 3, H (x) n =, x S. (1.6) 2 Transformation of the problem and formulation of main result. Lagrangian coordinates ξ 1 : x = ξ + t u(ξ, τ)dτ X(ξ, t), X : 1 1t, Γ Γ t. u(ξ, t) = v(x(ξ, t), t), u (x, t) : the extension of u from 1 in such that u (ξ, t) = in 3 and near S 3 S. x = ξ + t u (ξ, τ)dτ X(ξ, t), X : i it, i = 1, 2, 3 3, ( x ) L(u), L(u) = detl, L = LL 1 ; ξ 2

3 L = 1 for ξ 1, L T A(u): co-factors matrix, T means transposition. Let P = L T L/L, q(ξ, t) = p(x, t), h(ξ, t) = LH(X, t), e(ξ, t) = LE(X, t), i.e., H(X, t) = L L h, E(X, t) = L L e. The mapping X converts (1.1)-(1.5) in { ut ν 2 uu + u q u T M (Lh) = f(x, t), u u =, ξ 1, t >, µ(h t L T L t L h L T (u u ) L h) = rotp(ξ, t)e, L ξ i, i = 1, 2, 3, ProtPh = α(pe + µ(l 1 u h)), h =, ξ 1, rotph =, h =, e =, ξ 2, roth = αe + j(ξ, t), h =, ξ 3, (2.1) (2.2) { h n =, eτ =, ξ S, [µh n] =, [h τ ] =, [e τ ] = ξ S 3, [µh n ] =, [h τ ] = ( AT An An 2 n ))[h n ], [n Pe] = (u An )[µ]h τ, (T u (u, q) + [T M (Lh)])An =, ξ Γ, u(ξ, ) = v (ξ), ξ 1, h(ξ, ) = H (ξ), ξ i, i = 1, 2, 3, e(ξ, t) n(ξ)ds =, S k (2.3) (2.4) (2.5) where u = L T is the transformed gradient w.r.to x and is the gradient w.r.to ξ, n(x) = An (ξ)/ An (ξ), n : the exterior normal to Γ. T u = q + νs u (u): the transformed stress tensor; S u (w) = u w + ( u w) T : the transformed rate-of-strain tensor. We find the solution of (2.1)-(2.5) in anisotropic Sobolev-Slobodetskii spaces W r,r/2 2, r >. Theorem 1. Let Γ W l+3/2 2, u W2 l+1 ( 1 ), H W2 l( 1) W2 l( 2), W2 l( 3) with 3/2 < l < 2, and let the compatibility conditions (1.6) be satisfied. Assume also that f, f W l,l/2 2 (R 3 (, T )), j W (l 1)/2 2 (, T ; W 1 2 ( 3 )) L 2 (, T ; W l 2( 3 ), j(x, t) n(x) x S3 =. (2.6) 3

4 Then the problem (2.1)-(2.5) has a unique solution defined in a certain (small) time interval (, T ), T T with the following regularity properties: u W 2+l,1+l/2 2 (Q 1 T ), q W l,l/2 2 (Q 1 T ), q W l+1/2,l/2+1/4 2 (G T ), h W l+1,(l+1)/2 2 (Q i T ), e L 2 (, T ; W l 2( i )) W (l 1)/2 2 (, T ; W 1 2 ( i )), where i = 1, 2, 3,, Q i T = (, T ), G T = Γ (, T ). The solution satisfies the inequality u l+2,l/2+1 W 2 (Q 1 T ) + q W l,l/2 2 (Q 1 T ) + q l+1/2,l/2+1/4 W 2 (G T ) ( + h W l+1,l/2+1/2 2 (Q i T ) + e L 2 (,T ;W2 l( i)) ) + e (l 1)/2 W ( c + 2 (,T ;W 1 2 (Γ )) f W l,l/2 2 (Q 1 T ) + u W l+1 2 ( 1 ) H W l 2 ( i ) + j L 2 (,T ;W2 l( 3)) ) + j (l 1)/2 W 2 (,T ;W2 1(. 3)) (2.7) Exclusion of e. Let the test function ψ(ξ) satisfy the conditions ψ(ξ) =, ξ i, i = 1, 2, 3, rotψ(ξ) =, ξ 2, [µψ n ] =, [ψ τ ] =, ξ Γ, [µψ n] =, [ψ τ ] =, ξ S 3, ψ n =, ξ S. (2.8) Using the relation rotpe ψdξ = we write (2.1)-(2.5) as a nonlinear problem for u, q, h: Pe rotψdξ + [n Pe] ψds Γ 4

5 u t ν 2 uu + u q u T M (Lh) = f(x, t), u u =, ξ 1, t >, µ(h t Φ(h, u)) ψ(ξ, t)dξdt T + α 1 ProtPh rotψ(ξ, t)dξdt 1 3 µ 1 (L 1 u h) rotψ(ξ, t)dξdt 1 + Ψ(h, u) ψdsdt Γ = α 1 j(ξ, t) ψ(ξ, t)dξdt, 3 rotph =, ξ 2, h =, ξ 1 2 3, h n =, ξ S, [µh n] =, [h τ ] =, ξ S 3, (2.9) [µh n ] =, [h τ ] = ( AT An An 2 n ))[h n )] =, (T u (u, q) + [T M (Lh)])An =, ξ Γ, u(ξ, ) = v (ξ), ξ 1, h(ξ, ) = H (ξ), ξ i, i = 1, 2, 3, where Φ = L T L t L h + L T (u u ) L L h, Ψ = (u An )[µ]h τ. Theorem 2. Under the assumptions of Theorem 1, the problem (2.9) has a unique solution (u, q, h) satisfying the inequality u l+2,l/2+1 W 2 (Q 1 T ) + q W l,l/2 2 (Q 1 T ) + q l+1/2,l/2+1/4 W 2 (G T ) ( + h W l+1,l/2+1/2 2 (Q i T ) ( c + f W l,l/2 2 (Q 1 T ) + u W l+1 2 ( 1 ) H W l 2 ( i ) + j L 2 (,T ;W2 l( 3)) ) + j (l 1)/2 W 2 (,T ;W2 1(. 3)) (2.1) 5

6 3 Linear problems The proof of Theorem 2 is based on the analysis of some linear problems, namely, v t ν 2 v + p = f(ξ, t), v = f, ξ 1, t >, T (v, p)n = d(ξ, t), ξ Γ, v(ξ, ) = v (ξ), ξ 1, { roth(ξ) = k(x), h(ξ) =, ξ 1 2 3, [µh n] =, [h τ ] =, ξ Γ S 3, h n ξ S =, µh t ψ(ξ, t)dξdt + α 1 roth rotψ(ξ, t)dξdt 1 3 = g 1 (ξ, t) rotψ(y, t)dξdt µg 2 (y, t) ψ(ξ, t)dξdt, h(ξ, t) =, ξ 1 2 3, roth(ξ, t) =, ξ 2, [µh n ] =, [h τ ] =, ξ Γ, [µh n] =, [h τ ] =, ξ S 3, h n =, ξ S, h(ξ, ) = h (ξ), ξ 1 2 3, where ψ(ξ, t) is an arbitrary test vector field satisfying the conditions (2.8). Theorem 3. Assume that Γ W l+3/2 2, 3/2 < l < 2, and that the compatibility conditions f W l,l/2 2 (Q 1 T ), v W l+1 2 ( 1 ), f = F, f L 2 (, T ; W l+1 2 ( 1 )), F t W l/2 2 (, T ; L 2 ( 1 )), d W l+1/2,l/2+1/4 2 (G T ), (3.1) (3.2) (3.3) v (ξ) = f(ξ, ), (S(v )n) τ = d τ, ξ Γ are satisfied. Then the problem (3.1) has a unique solution v W 2+l,1+l/2 2 (Q 1 T ), p W l,l/2 1 (Q 1 T ) with p ξ Γ W l+1/2,l/2+1/4 2 (G T ) and v l+2,l/2+1 W 2 (Q 1 T ) + p W l,l/2 2 (Q 1 T ) + p W l+l/2,1/2+l/4 2 (Γ ) c( f L2 (,T ;W l+1 2 ( 1 )) + F t l/2 W 2 (,T ;L 2 ( 1 )) + d l+1/2,l/2+1/4 W 2 (G T ) + v W l+1 2 ( ) ). (3.4) 6

7 Let H l () be the space of vector fields ψ W2 l (), i = 1, 2, 3, satisfying (2.8); U(): finite dimensional set of vector fields satisfying (2.8) and, in addition, rotψ(ξ) =, ξ ; H l (): the subset of Hl () whose elements satisfy the orthogonality condition µψ(ξ) u(ξ)dξ =, u U(). Theorem 4. Let Γ, S, S 3 W l+3/2 2, l (3/2, 2). For arbitrary k W r 2 ( i), r [, l] i = 1, 2, 3, such that k =, ξ 1 2 3, [k n] =, ξ Γ S 3, k n S = the problem (3.2) has a unique solution belonging to W r+1 2 ( i ), i = 1, 2, 3 and orthogonal to U(). The solution satisfies the inequality then Moreover, if h W 1+r 2 ( i ) c k W r 2 ( i ). (3.5) k = rotk(ξ), [K τ ] Γ S 3 =, K τ S =, (3.6) h L2 () c K L2 (). (3.7) Corollary 1. If h H r+1 (), then c 1 h W r+1 2 ( i ) roth W2 r( 1 3 ) c 2 h W r+1 2 ( i ). Corollary 2. If k W r,r/2 2 (Q i T ), i = 1, 2, 3, then h L 2(, T : W2 r+1 ( i )) W r/2 2 (, T ; W2 1( i)) and ( h L2 (,T :W r+1 2 ( i )) + h W r/2 2 (,T ;W2 1( i)) ) (3.8) Moreover, if (3.6) holds, then c k W r,r/2 2 ( i ). h W (r+1)/2 2 (,T ;L 2 ()) c K W (r+1)/2 2 (,T ;L 2 ()). (3.9) Theorem 5. Let l (3/2, 2). For arbitrary h, g 1, g 2 satisfying the conditions h W l 2( i ), i = 1, 2, 3, h (y) =, y 1 2 3, roth (y) =, y 2, [µh n] =, [h τ ] =, y Γ S 3, h n =, y S, 7 (3.1)

8 g 1 L 2 (, T ; W l 2( 1 3 )) W (l 1)/2 2 (, T ; W 1 2 ( 1 3 )), g 1 =, g 1 n =, y Γ S 3, g 2 W l 1,(l 1)/2 2 (Q i T ), i = 1, 2, 3, [µg 2 n] =, [g 2τ ] =, y Γ S 3, g 2 n =, y S the problem (3.3) has a unique solution h W 1+l,(1+l)/2 2 (Q i T ), satisfying the inequality (3.11) h W 1+l,(1+l)/2 2 (Q i T ) c( ( h W l 2 ( i ) + g 2 l 1,(l 1)/2 W 2 (Q i )) T + ( g 1 L2 (,T ;W2 l( i)),3 + g 1 W (l 1)/2 2 (,T ;W 1 2 ( i) )), (3.12) and Moreover, there exist e L 2 (, T ; W2 l( i)) W (l 1)/2 2 (, T ; W2 1( i)), i = 1, 2, 3, such that µ(h t g 2 ) = rote(ξ, t), ξ 1 2 3, roth = α(e + g 1 ). ξ 1 3, (3.13) e(ξ, t) =, ξ 2, [e τ ] =, ξ Γ S 3, e τ =, ξ S, ( e L2 (,T ;W l 2 ( i)) + e W (l 1)/2 2 (,T ;W 1 2 ( i)) ) c( ( h W l 2 ( i ) + g 2 l 1,(l 1)/2 W 2 (Q i )) T + ( g 1 L2 (,T ;W2 l( i)) + g 1 (l 1)/2 W 2 (,T ;W2 1( i) )).,2 (3.14) We describe briefly the (formal) construction of e. We decompose h, h, ψ, g 2 in the sums h = h + h, h = h + h, ψ = ψ + ψ, g 2 = g 2 + g 2, where h, h, ψ, g 2 U() and h, h, ψ, g 2 are orthogonal to U(). It is easily verified that in the case h = h, g 2 = g 2 the problem (3.3) reduces to µ(h t g 2) ψ dξdt =, 8

9 i.e., and in the case h = h, h t g 2 =, h t= = h, (3.15) g 2 = g 2 we have h = h [1]. Let E be the solution of the problem rote = µ(h t g 2), E =, ξ, E τ =, ξ S. (3.16) It is easily seen that T 1 3 ( E + α 1 roth g 1 ) rotψ (y, t)dξdt =, which implies E + α 1 roth g 1 = χ 1 (ξ, t), ξ 1 3, (3.17) in view of Theorem 4. We set e = E + χ 1 (ξ, t), ξ 1 3, b e = E + χ 2 (ξ, t) + C j (t)v k (ξ), ξ 2, k=1 (3.18) where χ 2 is a solution of the problem 2 χ 2 (ξ, t) =, ξ 2, χ 2 (ξ, t) = χ 1 (ξ, t), ξ Γ S 3, χ 2 (ξ, t) =, ξ S (3.19) and v k are the Dirichlet vector fields in 2, i.e., v k = φ(ξ, t), 2 φ k (ξ, t) =, ξ 2, φ k (ξ, t) = δ jk, ξ S j, φ k (ξ, t) =, ξ S, (3.2) S j are all the connected components of 2, except S, j, k = 1,..., b, b is the second Betti number of the domain 2. The coefficients C j (t) may be found from the additional normalization conditions S k e nds =, that gives = (E + χ 2 (ξ, t) + S k b C j (t)v k (ξ)) nds. (3.21) The estimate (3.14) follows from (3.21)and from the estimates of solutions of (3.15), (3.16), (3.19), (3.2). k=1 9

10 4 Nonlinear problem We go back to Sec. 2 and write our main nonlinear problem in the form u t ν 2 u + q = f(ξ, t) + l 1 (u, q), u = l 2 (u), ξ 1, t >, T (u, q)n = l 3 (u), ξ Γ, µh t ψ(y, t)dydt + α 1 roth rotψ(y, t)dydt 1 3 = l 4 (u, h) rotψ(y, t)dydt µl 5 (u, h) ψ(y, t)dydt, roth = l 6 (u, h), ξ 2, [µh n] =, [h τ ] = l 7 (u, h), ξ Γ, [µh n] =, [h τ ] =, ξ S 3, h n =, ξ S, u(ξ, ) = v (ξ), ξ 1, h(ξ, ) = H (ξ), ξ i, i = 1, 2, 3, (4.1) where l 1 = ν( 2 u 2 )u + ( u )q + u T M (Lh) f(x, t) + f(ξ, t), l 2 = ( u ) h = (I A T )h, l 3 = Π (Π S(u)n ΠS u (u)n)) + νn ((n S(u)n ) (n S u (u)n)) + n [n T M (Lh)n], l 4 = α 1 (roth ProtPh) + µ(l 1 u h) (n Ψ ), l 5 = Φ(u, h) + µ 1 rot(n Ψ ), l 6 = rot(i P)h, (4.2) l 7 = ( AT An An 2 n )[h n ], where n and Ψ are extensions of n, Ψ from Γ in 1, Πf = f n(n f), Π f = f n (n f). 1

11 We solve the problem (4.1) by successive approximations, according to the scheme u m+1,t ν 2 u m+1 + q m+1 = f(ξ, t) + l 1 (u m, q m ), u m+1 = l 2 (u m ), ξ 1, t >, T (u m+1, q m+1 )n = l 3 (u m, h m ), ξ Γ, µh m+1t ψ(y, t)dydt + α 1 roth m+1 rotψ(y, t)dydt 1 3 = l 4 (u m, h m ) rotψ(y, t)dydt µl 5 (u m, h m ) ψ(y, t)dydt, roth m+1 = l 6 (u m, h m ), ξ 2, [µh m+1 n] =, [h m+1,τ ] = l 7 (u m, h m ), ξ Γ, [µh m+1 n] =, [h m+1,τ ] =, ξ S 3, h m+1 n =, ξ S, u m+1 (ξ, ) = v (ξ), ξ 1, h m+1 (ξ, ) = H (ξ), ξ i, i = 1, 2, 3, (4.3) u =, q =, ρ =, h =. Using Theorems 3-5 and estimates of the nonlinear terms (they are omitted), we obtain where β >, Y m+1 (T ) c 1 T β Ym(t) j + c 2 N(T ), (4.4) j=1 Y m (T ) = u m l+2,l/2+1 W 2 (Q 1 T ) + q m l,l/2 W 2 (Q 1 T ) + q m l+1/2,l/2+1/4 W 2 (G T ) + h m l+1,l/2+1/2 W 2 (Q i ), T N(T ) = f l,l/2 W 2 (WT i ) + j L 2 (,T ;W2 l( 3)) + j (l 1)/2 W 2 (,T ;W2 1( 3)) + v W l+1 2 ( 1 ) + h W l 2 ( i ). It follows from (4.4) that in the case of small T Y m+1 (T ) 2c 2 N(T ). (4.5) 11

12 The proof of the convergence of the sequence (u m, q m, ρ m, h m ) is based on the estimate of the differences (u m+1 u m, q m+1 q m, ρ m+1 ρ m, h m+1 ρ m ), as in [2]. This yields the proof of Theorem 2. The reconstruction of e can be made in the same manner as in the linear case above, which completes the proof of Theorem 1. References 1. S.Mosconi, V.A.Solonnikov, On a problem of magnetohydrodynamics in a multiconnected domain, Nonlinear Anal., 74 No 2 (211), M.Padula, V.A.Solonnikov, On the free boundary problem of magnetohydrodynamics, Zapiski Nauchn. Semin. POMI 385 (21), G.Stroemer, About the equations of non-stationary magnetohydrodynamics, Nonlin. Anal. 52 (23),