Abstract
We consider the motion of an incompressible magnetohydrodynamics with resistivity in a domain bounded by a free surface which is coupled through the free surface with an electromagnetic field generated by a magnetic field prescribed on an exterior fixed boundary. On the free surface, transmission conditions for the electromagnetic field are imposed. As transmission condition we assume jumps of tangent components of magnetic and electric fields on the free surface. We prove local existence of solutions such that velocity and magnetic fields belong to \(H^{2+\alpha ,1+\alpha /2}\), \(\alpha >5/8\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
We consider a free boundary problem for a viscous incompressible magnetohydrodynamic motion in domain \({\mathop {\Omega }\limits ^{1}}_t\) bounded by a free surface \(S_t\). The motion interacts with an electromagnetic field located in \({\mathop {\Omega }\limits ^{2}}_t\) (Fig. 1).
In \({\mathop {\Omega }\limits ^{1}}_t\) the magnetohydrodynamic motion is described by the following system of equations
where \(v=v(x,t)=(v_1(x,t),v_2(x,t),v_3(x,t))\in {\mathbb {R}}^3\) is the velocity of the fluid, \(p=p(x,t)\in {\mathbb {R}}\) is the pressure,\({\mathop {H}\limits ^{1}}={\mathop {H}\limits ^{1}}(x,t)=({\mathop {H}\limits ^{1}}_1(x,t),{\mathop {H}\limits ^{1}}_2(x,t), {\mathop {H}\limits ^{1}}_3(x,t)\in {\mathbb {R}}^3\) is the magnetic field, \({\mathop {E}\limits ^{1}}={\mathop {E}\limits ^{1}}(x,t)=({\mathop {E}\limits ^{1}}_1(x,t),{\mathop {E}\limits ^{1}}_2(x,t), {\mathop {E}\limits ^{1}}_3(x,t))\in {\mathbb {R}}^3\) is the electric field, \(f=f(x,t)=(f_1(x,t),f_2(x,t),f_3(x,t))\in {\mathbb {R}}^3\) is the external force field, and \(x=(x_1,x_2,x_3)\) are Cartesian coordinates. Moreover, \(\mu _1\) is the constant magnetic permeability, \(\sigma _1\) is the constant electric conductivity and \({\mathbb {T}}(v,p)\) is the stress tensor of the form
where \(\nu \) is the positive viscosity coefficient, \({\mathbb {I}}\) is the unit matrix and \({\mathbb {D}}(v)\) is the dilatation tensor of the form
For system (1.1) the following initial and boundary conditions are prescribed
where \({\bar{n}}\) is the unit vector outward to \({\mathop {\Omega }\limits ^{1}}_t\) and normal to \(S_t\), the constant exterior pressure \(p_0\) can be absorbed by pressure p and
The two dimensional cross section by a plane of the three-dimensional geometrical description of the considered free boundary problem
The boundary conditions \((1.4)_1\) implies the compatibility condition
where \({\bar{n}}(0)={\bar{n}}|_{t=0}\), \({{\bar{\tau }}}(0)={{\bar{\tau }}}|_{t=0}\) and \({{\bar{\tau }}}\) is a tangent vector to \(S_t\).
In \({\mathop {\Omega }\limits ^{2}}_t\) we have a motionless dielectric gas under a constant pressure \(p_0\). Therefore, we only have an electromagnetic field described by the system of equations
For system (1.6) the following initial and boundary conditions are prescribed:
where \({{\bar{\tau }}}_\alpha \), \(\alpha =1,2\), are tangent vectors to B.
The magnetohydrodynamic system (1.1) is composed of two problems. For a given magnetic field \({\mathop {H}\limits ^{1}}\) system \((1.1)_{1,2}\) determines velocity v and pressure p under appropriate initial and boundary conditons (1.4). To formulate a problem for \({\mathop {H}\limits ^{1}}\) we have to recall that a motion in \({\mathop {\Omega }\limits ^{1}}_t\) interacts with an electro-magnetic field in \({\mathop {\Omega }\limits ^{2}}_t\) through the free surface \(S_t\) by transmission conditions. Therefore, system (1.1\(_{3,4,5})\) and (1.6) is a problem for \({\mathop {H}\limits ^{1}}\) and \({\mathop {H}\limits ^{2}}\) which are coupled by transmission conditions on the interface \(S_t\). Therefore, for a given v, we have the following problem for \({\mathop {H}\limits ^{1}}\) and \({\mathop {H}\limits ^{2}}\):
where
and \({{\bar{\tau }}}_\alpha \), \(\alpha =1,2\), are tangent vectors to B. Electric vectors \({\mathop {E}\limits ^{1}}\) and \({\mathop {E}\limits ^{2}}\) are auxiliary.
To make system (1.8) complete we have to add transmission conditions. The conditions are recessary for a proof of existence of solutions to (1.1)–(1.8). Satisfying them we can derive such energy equality which can imply all necessary estimates for the proof of existence in the \(L_2\)-approach.
Lemma 1.1
Assume that near any point of \(S_t\) there exists an orthonormal system \(({{\bar{\tau }}}_1,{{\bar{\tau }}}_2,{\bar{n}})\), where \({{\bar{\tau }}}_\alpha \), \(\alpha =1,2\), is tangent to \(S_t\) and \({\bar{n}}\) is normal.
Let \(a_i\), \(i=1,2\), be given positive numbers. Let \(H_{*\alpha }=0\), \(\alpha =1,2\). Assume that
Then the following energy equality
holds.
Proof
From \((1.8)_1\) and \((1.8)_4\) we have
Recall the identity
where \(S_t=\partial \Omega _t\), \(\psi \) is a sufficiently regular function and \({\bar{n}}\) is the unit exterior vector to \(\Omega _t\) and normal to \(S_t\).
From (1.12) we have
where \({\mathop {{\bar{n}}}\limits ^{i}}\) is exterior to \({\mathop {\Omega }\limits ^{i}}_t\) and \({\mathop {{\bar{n}}}\limits ^{2}}=-{\mathop {{\bar{n}}}\limits ^{1}}\).
Using (1.13) and (1.14) in (1.11) yields
where \({\bar{n}}={\mathop {{\bar{n}}}\limits ^{1}}\). Using the orthonormal system \(({{\bar{\tau }}}_1,{{\bar{\tau }}}_2,{\bar{n}})\) we have
In view of (1.16) the boundary term in (1.15) equals
Hence (1.9) implies (1.10) and concludes the proof. \(\square \)
Remark 1.2
The boundary term in (1.15) needs more regularity than the second term. If the transmission condition does not hold equality (1.15) does not imply any estimate.
Remark 1.3
(Some discussion of transmission conditions can be found in [1]). There is many different transmission conditions.
-
1.
Let \(a_1^{\nu _1}{\mathop {E}\limits ^{1}}\cdot {{\bar{\tau }}}_\alpha =a_2^{\nu _2}{\mathop {E}\limits ^{2}}\cdot {{\bar{\tau }}}_\alpha ,\ \ a_1^{\nu _2}{\bar{n}}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{1}}=a_2^{\nu _2}{\bar{n}}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{2}},\ \ \alpha =1,2\), where \(\nu _1+\nu _2=1\) and \(a_1\), \(a_2\) are positive numbers. Then (1.9) holds.
-
2.
Let \(a_1=a_2=1\), \({\mathop {E}\limits ^{1}}\cdot {{\bar{\tau }}}_\alpha ={\mathop {E}\limits ^{2}}\cdot {{\bar{\tau }}}_\alpha \), \({\bar{n}}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{1}}={\bar{n}}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{2}}\), \(\alpha =1,2\). Then (1.9) also holds.
-
3.
Let \(a_1=a_2=1\), \({\mathop {E}\limits ^{1}}\cdot {{\bar{\tau }}}_\alpha =a{\mathop {E}\limits ^{2}}\cdot {{\bar{\tau }}}_\alpha \), \({\bar{n}}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{1}}={1\over a}\bar{n}\times {{\bar{\tau }}}_\alpha \cdot {\mathop {H}\limits ^{2}}\), \(\alpha =1,2\), where a is an arbitrary positive number. Then (1.9) is satisfied too.
-
4.
We can also define anisotropic transmission conditions.
In cases 1 and 3 we have jumps of tangent components of electric and magnetic fields.
We have to recall that in magnetohydrodynamics the displacement current \(E_{,t}\) is omitted.
To prove the existence of solutions to problem (1.1)–(1.9) we transform it into two problems: the problem for the fluid motion and the problem for the electromagnetic field. Therefore, for given \({\mathop {H}\limits ^{1}}\) we have the problem for (v, p):
where we assumed that \(p_0\) is absorbed by p.
Next for a given v, the electromagnetic field is determined by the problem:
Let v be a solution to (1.18). Then Lagrangian coordinates are the Cauchy data to the problem
Then
where \(\bar{v}(\xi ,t)=v(x(\xi ,t),t)\) describes a relation between Cartesian and Lagrangian coordinates.
Therefore, the transmission condition (1.9) holds on the surface
The structure of problems (1.18), (1.19) suggests an applying a method of successive approximations.
Eliminating the electric field in (1.19) yields
Now, we formulate the main result of this paper
Theorem 1.4
Assume that \(\Omega ={\mathop {\Omega }\limits ^{1}}_t\cup S_t\cup {\mathop {\Omega }\limits ^{2}}_t\). Assume that \(f\in H^{\alpha ,\alpha /2}(\Omega ^t)\), \(H_{*\beta }\in H^{3/2+\alpha ,3/4+\alpha /2}(B^t)\), \(S_0\in H^{3/2+\alpha }\), where \({5\over 8}<\alpha <1\), \(\beta =1,2\). Assume that \(v(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{1}}_0)\), \({\mathop {H}\limits ^{i}}(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{i}}_0)\), \(i=1,2\).
Then for T sufficiently small there exists a local solution to problem (1.18)–(1.20) such that \({\bar{v}}\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \({\mathop {{\bar{H}}}\limits ^{i}}\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{i}}_0^t)\), \(i=1,2\), \({\bar{p}}_\xi ,{\bar{p}}\in L_2({\mathop {\Omega }\limits ^{1}}_0^t)\), \({\bar{p}}|_{S_0}\in H^{1/2+\alpha ,1/4+\alpha /2}(S_0^t)\) and the estimate
holds, where \(t\le T\) and \({\bar{v}}\), \({\mathop {{\bar{H}}}\limits ^{i}}\), \({\bar{p}}\), \({\bar{f}}\) are equal to v, \({\mathop {H}\limits ^{i}}\), p, f expressed in Lagrangian coordinates. Moreover, the Sobolev–Slobodetskii space \(H^{k+\alpha ,k/2+\alpha /2}\!\), \(k\in {\mathbb {N}}\), \(\alpha \in (0,1)\) is defined in (2.2.1).
In this paper we prove existence of local solutions to problem (1.18), (1.20). The formulation of problem (1.18), (1.20) suggests that the method of successive approximations should be used. The method is described in Sect. 3 in problems (3.8) and (3.9). The Stokes system (3.8) with the Neumann boundary conditions determines a relation between \(v_{n+1}\), \(p_{n+1}\) at the \(n+1\)-th step of iteration and \(v_n\), \(p_n\), \({\mathop {H}\limits ^{1}}_n\) at the n-th step. Similarly, \({\mathop {H}\limits ^{1}}_{n+1}\), \({\mathop {H}\limits ^{2}}_{n+1}\) are solutions to problem (3.9) for given \(v_n\) \({\mathop {H}\limits ^{1}}_n\), \({\mathop {H}\limits ^{2}}_n\). The existence of solutions to problem (3.8) follows from Lemma 2.3.2. Problem (3.9) describes \({\mathop {H}\limits ^{1}}_{n+1}\) and \({\mathop {H}\limits ^{2}}_{n+1}\) in domains \({\mathop {\Omega }\limits ^{1}}_0\) and \({\mathop {\Omega }\limits ^{2}}_0\), respectively. Moreover, \({\mathop {H}\limits ^{1}}_{n+1}\) and \({\mathop {H}\limits ^{2}}_{n+1}\) are coupled through the free surface \(S_0\) by transmission conditions (see Lemma 1.1) and \({\mathop {H}\limits ^{2}}_{n+1}\) satisfies some boundary conditions on B. Existence of solutions to problem (3.9) follows from Lemma 2.5.1. The proof of this lemma exploits the technique of regularizer introduced in [2]. Since Lagrangian coordinates are used domains \({\mathop {\Omega }\limits ^{1}}_0\), \({\mathop {\Omega }\limits ^{2}}_0\) are the initial domains and \(S_0\) is the initial free boundary.
Exploiting Lagrangian coordinates the r.h.s. of problems (3.8) and (3.9) are strongly nonlinear and complicated.
In Sect. 4 and 5 we derive the inequality (see Corollaries 4.5 and 5.4)
where
where \(V_2^{2+\alpha }(\Omega ^t)\) is defined by
\(\alpha >5/8\), \(\phi \) is a strongly nonlinear increasing function and \(a>0\).
Thanks to the coefficient \(t^a\) and t sufficiently small we derive the estimate (see Lemma 6.1)
Using differences introduced in Lemma 6.2 we prove in Sect. 6 that the sequence \(({\bar{v}}_n,{\mathop {{\bar{H}}}\limits ^{1}}_n,{\mathop {{\bar{H}}}\limits ^{2}}_n)\) converges. Hence, we prove the existence of solutions to problem (1.18), (1.20).
We should explain some points of the proof of Lemma 2.5.1 which needs the technique of regularizer. This technique needs to examine the following local problems derived from (3.9):
-
1.
near an interior point of \({\mathop {\Omega }\limits ^{1}}_0\),
-
2.
near an interior point of \({\mathop {\Omega }\limits ^{2}}_0\),
-
3.
near a point of \(S_0\),
-
4.
near a point of B.
The local problems in cases 3 and 4 are considered in Sects. 7 and 8, respectively. To solve the local problems we need Besov spaces \(H^{2+\alpha ,1+\alpha /2}\) expressed in the Fourier-Laplace transforms. This is possible because \(H^{2+\alpha ,1+\alpha /2}\) are \(L_2\)-Besov spaces.
The equations of magnetohydrodynamics (mhd) can be found in [3, 4].
The first result on solvability of mhd equations appeared in [5]. Later free boundary problems to incompressible viscous mhd with resistivity were considered in [6].
Free boundary problems for mhd equations were also considered in [7, 8]. In [7, 8] the external magnetic field satisfies the elliptic system
However in those papers the boundary condition on the free surface contains the surface tension.
The existence of local solutions to problem (1.18), (1.20) has been already considered in [2, 9, 10]. Comparing to [2] in this paper we proved existence of solutions with the lowest possible regularity. Moreover, we used the \(L_2\)-approach because we are going to prove global existence of solutions using appropriate differential inequalities (see [11]). In [9, 10] the existence of solutions to problem (1.18), (1.20) is proved by the Faedo–Galerkin method. The applied energy method to solutions to problem (1.20) in [9, 10] implies very strong restrictions on the transmission coefficients. In this approach the proof of existence with optimal regularity is not possible.
In this paper we express the results in [9] in a more explicit and appropriate way.
In [12, 13], using the Weis theory of Fourier multipliers, two different mhd fluids inteacting through a free surface are considered.
The result similar to Lemma 2.5.1 is also proved in [14].
In [15,16,17] the global existence of solutions to problem (1.18), (1.20) is proved by the energy method so only Sobolev spaces are used.
Moreover, the methods used in [15,16,17] imposes strong restrictions on the transmission coefficients. In the forthcoming paper we will relax the restrictions.
2 Notation and Auxiliary Results
2.1 Partition of Unity
To prove the existence of solutions to problem (1.18), (1.20) we need a partition of unity. We consider two collections of open subsets \(\{\omega ^{(k)}\}\) and \(\{\Omega ^{(k)}\}\), \(k\in {\mathfrak {M}}\cup {\mathfrak {N}}\), such that \({{\bar{\omega }}}^{(k)}\subset \Omega ^{(k)}\subset \Omega _0={\mathop {\Omega }\limits ^{1}}_0\cup {\mathop {\Omega }\limits ^{2}}_0\), \(\bigcup _k\omega ^{(k)}=\bigcup _k\Omega ^{(k)}=\Omega _0\), \({{\bar{\Omega }}}^{(k)}\cap S_0=\phi _0\), where \(\phi _0\) describes the empty set for \(k\in {\mathfrak {M}}_1\cup {\mathfrak {M}}_2\), \({{\bar{\Omega }}}^{(k)}\cap S_0\not =\phi _0\), where \(\phi _0\) describes the empty set for \(k\in {\mathfrak {N}}_1\) and \({{\bar{\Omega }}}^{(k)}\cap B\not =\phi _0\), where \(\phi _0\) describes the emnpty set for \(k\in {\mathfrak {N}}_2\), \({\mathfrak {N}}={\mathfrak {N}}_1\cup {\mathfrak {N}}_2\). Moreover, subdomains with \(k\in {\mathfrak {M}}_i\) are contained in \({\mathop {\Omega }\limits ^{i}}_0\), \(i=1,2\).
We assume that at most \(N_0\) of the \(\Omega ^{(k)}\) have nonempty intersections, \(\sup _k\textrm{diam}\,\Omega ^{(k)}\le 2\lambda \), \(\sup _k \textrm{diam}\,\omega ^{(k)}\le \lambda \) for some \(\lambda >0\). Let \(\zeta ^{(k)}(x)\) be a smooth function such that \(0\le \zeta ^{(k)}(x)\le 1\), \(\zeta ^{(k)}(x)=1\) for \(x\in \omega ^{(k)}\), \(\zeta ^{(k)}(x)=0\) for \(x\in \Omega _0\setminus \Omega ^{(k)}\) and \(|D_x^\nu \zeta ^{(k)}(x)|\le c/\lambda ^{|\nu |}\). Then \(1\le \sum _k(\zeta ^{(k)}(x))^2\le N_0\). Introducing the function
we have that \(\eta ^{(k)}(x)=0\) for \(x\in \Omega _0\setminus \Omega ^{(k)}\), \(\sum _k\eta ^{(k)}(x)\zeta ^{(k)}(x)=1\) and \(|D_x^\nu \eta ^{(k)}(x)|\le c/\lambda ^{|\nu |}\).
We denote by \(\xi ^{(k)}\) an interior point of \(\omega ^{(k)}\) and \(\Omega ^{(k)}\) for \(k\in {\mathfrak {M}}\) and an interior point of \({{\bar{\omega }}}^{(k)}\cap S_0\) and of \({{\bar{\Omega }}}^{(k)}\cap S_0\) for \(k\in {\mathfrak {N}}_1\) and an interior point of \({{\bar{\omega }}}^{(k)}\cap B\) and of \({{\bar{\Omega }}}^{(k)}\cap B\) for \(k\in {\mathfrak {N}}_2\). For \(k\in {\mathfrak {M}}_i\), \(\xi ^{(k)}\in \Omega _0^i\), \(i=1,2\). Let \(x=(x_1,x_2,x_3)\) be the Cartesian system of coordinates with the origin located in the interior of \(\Omega _0\).
Then by translations and rotations we introduce a local coordinate system \(y=(y_1,y_2,y_3)\) with the origin at \(\xi ^{(k)}\in \Omega ^{(k)}\cap S_0\), \(k\in {\mathfrak {N}}_1\), such that the part \({\tilde{S}}_0^{(k)}=S_0\cap {{\bar{\Omega }}}^{(k)}\) of the boundary \(S_0\) is described by \(y_3=F_k(y_1,y_2)\). We denote the transformation as \(y=Y_k(x)\). Then we introduce new coordinates defined by
We will denote this transformation by \({{\hat{\Omega }}}^{(k)}\ni z=\Phi _k(y)\), where \(y\in \Omega ^{(k)}\).
We assume that the sets \({{\hat{\omega }}}^{(k)}\), \({{\hat{\Omega }}}^{(k)}\) are described in local coordinates at \(\xi ^{(k)}\) by the inequalities
respectively. Moreover, \((y_1,y_2,y_3)\in {\mathop {\Omega }\limits ^{1}}_0\) if \(y_3>F_k(y_1,y_2)\) and \((y_1,y_2,y_3)\in {\mathop {\Omega }\limits ^{2}}_0\) for \(y_3<F_k(y_1,y_2)\). Let \(\Psi _k=\Phi _k\circ Y_k\). Then \(z=\Psi _k(x)\) and
For \(k\in {\mathfrak {M}}\) we have
For \(k\in {\mathfrak {N}}_2\) we introduce new local coordinates with origin at \(\xi ^{(k)}\in B\cap {{\bar{\Omega }}}^{(k)}\) such that \(y_3=F_k(y_1,y_2)\) describes locally \(B\cap {{\bar{\Omega }}}^{(k)}\). We also introduce the transfomation \(z_i=y_i\), \(i=1,2\), \(z_3=y_3-F_k(y_1,y_2)\) and assume that \(z=\Phi _k(y)\) belongs to \({{\hat{\Omega }}}^{(k)}\) for \(y\in \Omega ^{(k)}\).
Finally, \({{\hat{\omega }}}^{(k)}\), \({{\hat{\Omega }}}^{(k)}\) are described by the inequalities
respectively.
Moreover, we introduce the notation: r.h.s. (l.h.s.) right-hand side (left-hand side).
By \(\phi \) we denote an increasing positive function such that \(\phi (0)\not =0\) and it can change its form from formula to formula.
2.2 Spaces
We prove the existence of local solutions to problem (1.18), (1.20) in \(L_2\)-Sobolev-Slobodetskii spaces with the norm
where \(\alpha \in (0,1)\), [l] is the integer part of l,
We need the Hilbert type spaces because their norms can be expressed in the Fourier-Laplace transforms. We also need the Hilbert type spaces because a proof of global existence of solutions to problem (1.18), (1.20) will be made by the energy method.
In this paper we also need the \(L_p\)-Besov spaces. Hence we recall some properties of isotropic Besov spaces which are frequently used in this paper. Next, we define anisotropic Besov spaces and formulate some imbedding theorems which we need.
Let us introduce the differences
where \(x\in {\mathbb {R}}^n\) and \(e_i\) \(i=1,\dots ,n\), are the standard unit vectors. Then we define inductively the m-difference
where \(c_{jm}={m\atopwithdelims ()j}={m!\over j!(m-j)!}\). Moreover, we introduce the difference
and inductively
Since
we have
where the last equality holds for \((x-y)\cdot e_i=h_i\).
We define the isotropic Besov spaces by introducing the norm (see [18, Ch. 4, Sect. 18])
where \(m>l-k\), \(m,k\in {\mathbb {N}}\cup \{0\}\), \(l\in {\mathbb {R}}_+\), \(l\not ={\mathbb {Z}}\), \(p\in (1,\infty )\).
It was shown in [19] that the Besov spaces defined by (2.2.2) all coincide and have equivalent norms for all m, k satisfying \(m>l-k\).
Next we define the \(L_p\)-scale of Sobolev-Slobodetskii spaces by introducing the norm
where \(l\not \in {\mathbb {Z}}\), [l] is the integer part of l.
We frequently write \(l=k+\alpha \), \(k\in {\mathbb {N}}\cup \{0\}\), \(\alpha \in (0,1)\), so \(k=[l]\) and \(\alpha =l-[l]\).
By the Golovkin theorem (see [19]) the norms of the spaces \(B_p^l({\mathbb {R}}^n)\) and \(W_p^l({\mathbb {R}}^n)\) are equivalent.
We also define the norms
for any \(m>l-k\) and
Now we introduce the partial derivatives
Let
We also need the following seminorms
and
where \(\Omega \subset {\mathbb {R}}^3\).
Lemma 2.2.1
The following imbeddings
where
where
and
where
hold.
We recall the following theorems of imbedding freguently used in the paper
Lemma 2.2.2
-
1.
Let \(l,l_1\in {\mathbb {R}}_+\), \(p,p_1\in (1,\infty )\), \(p_1\ge p\). Let \(\Omega \subset {\mathbb {R}}^3\). If \(3/p-3/p_1+l_1\le l\) then
$$\begin{aligned} W_p^l(\Omega )\subset W_{p_1}^{l_1}(\Omega ). \end{aligned}$$ -
2.
If \({3\over p}-{3\over q}+\alpha \le l\), \(\alpha \in {\mathbb {N}}\cup \{0\}\), \(q\in [1,\infty ]\).
Then
Consider anisotropic Sobolev-Slobodetskii spaces \(W_{p,q}^{l,l/2}(\Omega \times (0,T))\) with the norm
where [l] is the integer part of \(l,p,q\in [1,\infty ]\),
where \(\alpha \in (0,1)\) and
Lemma 2.2.3
Let \(l,l'\in {\mathbb {R}}_+\), \(p,q,p',q'\in [1,\infty ]\), \(p'\ge p,q'\ge q\), \(\Omega \subset {\mathbb {R}}^3\). Let
Then the imbedding
holds.
Now, we recall the trace theorems from [20].
Lemma 2.2.4
(Trace theorem) Let \(S=\partial \Omega \). Let \(u\in W_{p,q}^{l,l/2}(\Omega \times (0,T))\), \(l\in {\mathbb {R}}_+\), \((p,q)\in (1,\infty )\). Let \(\varphi =u|_{S^T}\) be the trace of u on \(S^T\). Then \(\varphi \in W_{p,q}^{l-1/p,l/2-1/2q}(S\times (0,T))\) and
where c does not depend on u.
Lemma 2.2.5
(Inverse trace theorem) Let \(\varphi \in W_{p,q}^{l-1/p,l/2-1/2q}(S\times (0,T))\), \(l\in {\mathbb {R}}_+\), \((p,q)\in (1,\infty )\). Then there exists a function \(u\in W_{p,q}^{l.l/2}(\Omega \times (0,T))\) such that \(u|_{S^T}=\varphi \) and there exists a constant c independent of \(\varphi \) such that
Lemma 2.2.6
(Time trace theorem) Let \(u\in W_{p,q}^{l,l/2}(\Omega \times (0,T))\), \(l\in {\mathbb {R}}_+\), \(p,q\in (1,\infty )\), \(t_0\in (0,T)\). Then the time trace \(\varphi =u|_{t=t_0}\) belongs to \(W_p^{l-2/q}(\Omega )\) and there exists a constant c independent of u such that
Lemma 2.2.7
(Inverse time trace theorem) Let \(\varphi \in W_p^{l-2/q}(\Omega )\), \(l\in {\mathbb {R}}_+\), \(p,q\in (1,\infty )\). Then there exists a function \(u\in W_{p,q}^{l,l/2}(\Omega \times (0,T))\) such that
and
Finally, we introduce the energy type space
2.3 The Stokes System
We consider the following Stokes problem in a bounded domain \(\Omega \) in \({\mathbb {R}}^3\) with boundary S,
Lemma 2.3.1
(see [21])
-
(a)
Assume that \(f\in H^{\alpha ,\alpha /2}(\Omega ^T)\), \(b\in H^{3/2+\alpha ,3/4+\alpha /2}(S^T)\), \(w_0\in H^{1+\alpha }(\Omega )\), \(\alpha \in (0,1)\). Then there exists a solution to problem (2.3.1) such that \(w\in H^{2+\alpha ,1+\alpha /2}(\Omega ^T)\), \(\nabla p\in H^{\alpha ,\alpha /2}(\Omega ^T)\) and there exists a function c(T, S) such that
$$\begin{aligned} \begin{aligned}&\Vert w\Vert _{H^{2+\alpha ,1+\alpha /2}(\Omega ^t)}+\Vert \nabla p\Vert _{H^{\alpha ,\alpha /2}(S^t)}\\&\quad \le c(T,S)(\Vert f\Vert _{H^{\alpha ,\alpha /2}(\Omega ^t)}+ \Vert b\Vert _{H^{3/2+\alpha ,3/4+\alpha /2}(S^t)}\\&\qquad +\Vert w_0\Vert _{H^{1+\alpha }(\Omega )}),\end{aligned} \end{aligned}$$(2.3.2)where \(t\le T\).
-
(b)
Assume that \(f\in L_2(\Omega ^T)\), \(b\in H^{3/2,3/4}(S^T)\), \(w_0\in H^1(\Omega )\). Then there exists a solution to problem (2.3.1) such that \(w\in H^{2,1}(\Omega ^T)\), \(\nabla p\in L_2(\Omega ^T)\) and there exists a function c(T, S) such that
$$\begin{aligned} \begin{aligned}&\Vert w\Vert _{H^{2,1}(\Omega ^t)}+\Vert \nabla p\Vert _{L_2(\Omega ^t)}\le c(T,S)(\Vert f\Vert _{L_2(\Omega ^t)}\\&\quad +\Vert b\Vert _{H^{3/2,3/4}(S^t)}+\Vert w_0\Vert _{H^1(\Omega )}),\end{aligned} \end{aligned}$$(2.3.3)where \(t\le T\).
Consider the Neumann problem to the Stokes system
To apply [21] we introduce a function \(\varphi \) satisfying the Dirichlet problem to the Laplace equation
Then we introduce the divergence free function
where w is a solution to the following problem
The existence of solutions to problem (2.3.7) is described by the following lemma
Lemma 2.3.2
Let T be a positive arbitrary finite number, \(S\in H^{3/2+\alpha }\), \(\alpha \in (1/2,1)\). Let \(f,\nabla g,\nabla \varphi _t\in H^{\alpha ,\alpha /2}(\Omega ^T)\), \(w_0\in H^{1+\alpha }(\Omega )\), \(h\in H^{1/2+\alpha ,1/4+\alpha /2}(S^T)\). Assume the compatibility conditions
where \({\bar{n}}\) and \({{\bar{\tau }}}_\beta \) are normal and tangent vectors to S.
Then there exists a unique solution to problem (2.3.7) such that \(w\in H^{2+\alpha ,1+\alpha /2}(\Omega ^t)\), \(p\in H^{\alpha ,\alpha /2}(\Omega ^t)\), \(\nabla p\in H^{\alpha ,\alpha /2}(\Omega ^t)\), \(p|_{S^t}\in H^{1/2+\alpha ,1/4+\alpha /2}(S^t)\) and the inequality holds
where \(t\le T\).
In view of (2.3.6) we can write (2.3.8) in the form
where H(t) is defined in (2.3.8).
Proofs of Lemmas 2.3.1 and 2.3.2 can be found in [21], where definitions of Besov spaces introduced in [22] were used.
Existence of solutions to the Stokes system can be also found in [23,24,25].
2.4 Transformation Between Eulerian and Lagrangian Coordinates
Let \(v=v(x,t)\) be given. Lagrangian coordinates are the Cauchy data for the problem
where \(x=(x_1,x_2,x_3)\) are Cartesian coordinates.
Integrating the above problem with respect to time yields
Then we define
Then the transformation between Cartesian coordinate x and Lagrangian coordinate \(\xi \) is described by the relation
The Jacobian of this transformation is the matrix
We have \(A^{-1}=\{\xi _{j,x_i}\}=\{a^{ji}\}\), \(\det A=\exp \intop _0^t\textrm{div}\,_uudt'=1\), \({\mathcal {A}}=(A^T)^{-1}\) is the matrix of cofactors. Denoting \({\mathcal {A}}=\{A_{ij}\}\) we have \(a^{mj}=A_{jm}\). Since incompressible motions are considered, we have \(\sum _kA_{ik,\xi _k}(\xi ,t)=0\) and \(\nabla _u={\mathcal {A}}\cdot \nabla _\xi =\nabla _\xi \cdot {\mathcal {A}}^T\).
Assume that \(S_t\) is given, at least locally, by the equation \(F(x)=0\) and \(S_0\) by \(F(\xi )=0\).
Then the normal vectors to \(S_t\) and \(S_0\) are given, respectively, by
Then
Since \(\xi _x=x_\xi ^{-1}\) we obtain
where \(\phi \) is an increasing positive function such that \(\phi (0)\not =0\). \(\phi \) will play a role of the generic function because it can change its form from formula to formula.
By imbedding we have
where \(\alpha >1/2\).
In view of definition of \(\delta _u(t)\) we have
Continuing, we have
2.5 Problem for the Magnetic Field
In this paper we restrict our considerations to transmission condition described in the case 2 of Remark 1.3. Then problem (1.20) takes the following form
Recall that problem (2.5.1) is formulated in Cartesian coordinates.
To examine problem (2.5.1) we have to consider first the following problem with constant coefficients formulated in Lagrangian coordinates
To prove the existence of solutions to problem (2.5.2) we exploit the technique of regularizer described in [26, Ch. 4, Sect. 7] and also utilized in [2, Sect. 5 and Sect. 10].
Introduce function \({\mathop {\Phi }\limits ^{i}}\), \(i=1,2\), as a solution to the problem
where \(\delta _{2i}\) is the Kronecker delta.
Lemma 2.5.1
Let \(\alpha \in (0,1)\) and let \(T>0\) be given. Assume that \({\mathop {M}\limits ^{i}}\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{i}}_0^t)\), \(\nabla {\mathop {{{\bar{\Phi }}}}\limits ^{i}}\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{i}}_0^t)\), \(\nabla {\mathop {N}\limits ^{i}}\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{i}}_0^t)\), \({\mathop {H}\limits ^{i}}(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{i}}_0)\), \(i=1,2\). Assume that \(H_{*j}\in H^{2+\alpha -1/2,1+\alpha /2-1/4}(B^t)\), \(B\in H^{1/2+\alpha ,1/4+\alpha /2}(B^t)\), \(K_j\in H^{1/2+\alpha ,1/4+\alpha /2}(S_0^t)\), \(L_j\in H^{3/2+\alpha ,3/4+\alpha /2}(S_0^t)\), \(j=1,2\), \(t\le T\). Then there exists a unique solution to problem (2.5.2) such that \({\mathop {{\bar{H}}}\limits ^{i}}\in V_2^{2+\alpha }({\mathop {\Omega }\limits ^{i}}_0^t)\), \(i=1,2\), and
where \(t\le T\).
Proof
The proof is divided into the following steps:
-
1.
First we introduce a partition of unity connected with the four kinds of subdomains:
- \(1_1\):
-
a neighborhood of an interior point of \({\mathop {\Omega }\limits ^{1}}_0\) located in a positive distance from \(S_0\),
- \(1_2\):
-
a neighborhood of an interior point of \({\mathop {\Omega }\limits ^{2}}_0\) located in a positive distance from \(S_0\) and B,
- \(1_3\):
-
a neighborhood of a point of \(S_0\),
- \(1_4\):
-
a neighborhood of a point of B.
There is constructed a partition of unity with supports corresponding to mentioned above subdomains.
-
2.
Using the partition of unity problem (2.5.2) is localized to the above neighborhoods. The local problems in subdomains \(1_1\) and \(1_2\) can be easily solved. The local problem in subdomain \(1_3\) is solved in Sect. 7 and in \(1_4\) in Sect. 8.
-
3.
To solve problem (2.5.2) we collect results of all above defined local problems by using the key idea of resularizer (see [26, Ch. 4]). Using the proofs of existence of solutions to problem (2.5.2) in Sects. 5 and 10 from [2] we conclude the proof of the lemma.
\(\square \)
The result similar to Lemma 2.5.1 is also proved in [14].
2.6 Spaces Defined by the Fourier–Laplace Transforms
In this subsection we follow [25]. Assume that \(u\in H^{\alpha ,\alpha /2}({\mathbb {R}}^3\times {\mathbb {R}}_+)\). Assume that u can be extended by zero for \(t<0\) and the extended \(u\in H^{\alpha ,\alpha /2}({\mathbb {R}}^3\times {\mathbb {R}})\).
We define the Fourier-Laplace transform for functions vanishing sufficiently fast at infinity by
where Re \(s>0\).
For any \(u\in H^{\alpha ,\alpha /2}({\mathbb {R}}^3\times {\mathbb {R}})\) the Fourier-Laplace transform is a holomorphic function for Re \(s>\gamma \), \(\gamma >0\). We introduce the norm
where \(s=\gamma +i\xi _0\).
Lemma 2.6.1
(see Lemma 2.1 from [25]). There exist constants \(c_1\), \(c_2\) such that
Finally, we recall
Lemma 2.6.2
(see Lemma 3.1 from [25]). Let \(e(x_3)=e^{-\tau x_3}\), \(\tau =\sigma s+\xi ^2\), \(s=\gamma +i\xi _0\), \(\gamma >0\). Then
3 Method of Successive Approximations
Let \(v=v(x,t)\) be given, where \(x\in {\mathop {\Omega }\limits ^{1}}_t\).
Definition 3.1
The Lagrangian coordinates in \({\mathop {\Omega }\limits ^{1}}_0\) are initial data to the Cauchy problem
Hence the domain \({\mathop {\Omega }\limits ^{1}}_t\) is defined by
where \({\bar{v}}(\xi ,t)=v(x(\xi ,t),t)\).
In free boundary problems in hydrodynamics the free boundary is built from the same fluid particles as at time \(t=0\) because \(v|_{S_t}\) is tangent to \(S_t\). Then
To formulate problem (2.5.1) in Lagrangian coordinates we have to introduce them in \({\mathop {\Omega }\limits ^{2}}_0\). Since there is no velocity in \({\mathop {\Omega }\limits ^{2}}_t\), we have to introduce it artificially.
Definition 3.2
Let \({\mathop {v}\limits ^{1}}=v\) in \({\mathop {\Omega }\limits ^{1}}_t\) and construct \({\mathop {v}\limits ^{2}}\) in \({\mathop {\Omega }\limits ^{2}}_t\) as a solution to the nonstationary Stokes problem
where q plays the role of pressure but it is not important for any estimate for \({\mathop {v}\limits ^{2}}\). It is introduced to have \({\mathop {v}\limits ^{2}}\) divergence free.
Finally, we construct \({\mathop {v}\limits ^{2}}(0)\) as a solution to the stationary Stokes system
Having \({\mathop {v}\limits ^{2}}\) constructed by problems (3.2) and (3.3), we can introduce Lagrangian coordinates \({\mathop {\xi }\limits ^{1}}\), \({\mathop {\xi }\limits ^{2}}\) by the Cauchy data to the problems
Then
where \({\mathop {{\bar{v}}}\limits ^{i}}({\mathop {\xi }\limits ^{i}},t)={\mathop {v}\limits ^{i}}({\mathop {x}\limits ^{i}}({\mathop {\xi }\limits ^{i}},t),t)\), \({\mathop {\xi }\limits ^{i}}\in {\mathop {\Omega }\limits ^{i}}_0\), \(i=1,2\).
Expressing problems (1.18) and (2.5.1) in Lagrangian coordinates yields
and
where \({\bar{v}}={\mathop {{\bar{v}}}\limits ^{1}}={\mathop {{\bar{v}}}\limits ^{2}}\) on \(S_0\) and \(\nabla _{{\bar{v}}}={\partial \xi _k\over \partial x}|_{x=x(\xi ,t)}\partial _{\xi _k}\).
Moreover, any operator with subscript \({\bar{v}}\) means that it contains the transformed gradient \(\nabla _{{\bar{v}}}\) and any operator with subscript \(\xi \) contains derivatives with respect to \(\xi \).
To prove existence of local solutions to problem (3.6), (3.7) we apply the following method of successive approximations
where \({\bar{v}}_n={\mathop {{\bar{v}}}\limits ^{1}}_n\), and
In problems (3.8) and (3.9) \({\mathop {{\bar{v}}}\limits ^{1}}_n\), \({\mathop {{\bar{v}}}\limits ^{2}}_n\), \({\mathop {{\bar{H}}}\limits ^{1}}_n\), \({\mathop {{\bar{H}}}\limits ^{2}}_n\) are treated as given.
Moreover, \({{\bar{\tau }}}_\alpha \), \(\alpha =1,2\), are tangent to \(S_0\), \({\bar{n}}\) is normal and \({{\bar{\tau }}}'_\alpha \), \(\alpha =1,2\), are tangent to B.
4 Estimates for Solutions to Problem (3.8)
Let \(\varphi _n\) be a solution to the problem
There exists the Green function to problem (4.1) such that
Lemma 2.3.2 yields
Lemma 4.1
Assume that \(F_n\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \(\nabla g_n\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \(G_n\in H^{1/2+\alpha ,1/4+\alpha /2}S_0^t)\), \(v(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{1}}_0)\), \(\nabla \varphi _n\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \(\nabla \varphi _t\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\). Then there exists a unique solution to problem (3.8) such that \({\bar{v}}_{n+1}\in V_2^{2+\alpha }({\mathop {\Omega }\limits ^{1}}_0^t)\), \(\nabla {\bar{p}}_{n+1},{\bar{p}}_{n+1}\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \({\bar{p}}_{n+1}\in H^{1/2+\alpha ,1/4+\alpha /2}(S_0^t)\) and the estimate holds
The expressions on the r.h.s. of (4.3) depend on \({\bar{v}}_n\), \({\bar{p}}_n\) and \({\mathop {{\bar{H}}}\limits ^{1}}_n\). Therefore we estimate them using the dependence.
Lemma 4.2
Assume that \({\bar{v}}_n,{\mathop {{\bar{H}}}\limits ^{1}}_n\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\), \({\bar{p}}_{n,\xi }\in H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\).
Assume that \(\alpha >5/8\), \(a>0\).
Then
where \(\delta _{{\bar{v}}_n}(t)\) is introduced in (2.4.2).
Proof
We have
In this proof we drop the index n and introduce the simplified notation
First we consider
The norm of space \(H^{\alpha ,\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)\) can be expressed in the form
Using Sect. 2.4 we have
The second term on the r.h.s. of (4.8) is bounded by
First, we examine
In view of properties of matrix \(\xi _x\), we have
The Hölder inequality implies
where \(1/p+1/p'=1\), \(p'<2\), \(\alpha '=\alpha +{1\over 2p}\big ({3\over 2}p-3\big )\).
By the Minkowski and Hölder inequalities we get
where the above imbedding holds under the condition
However \(p>2\) the last restriction can hold for p close to 2.
Since \(p'<2\) we obtain
where the last inequality holds for
Conditions (4.11) and (4.12) imply
Summarizing,
where we used that \(J_2\) is also bounded by the above bound.
Next, we examine
First, similarly as in the estimate of \(J_1^1\) we derive
Next
Finally, \(H_3\) is estimated by the same bound as \(H_1\).
Summarizing,
Finally, we estimate the last term on the r.h.s. of (4.10),
Using the form of \(\xi _x\), we have
\(M_1^1\) has the same structure as \(J_1^1\), where \(v_{\xi \xi }\) is replaced by \(p_\xi \). Therefore, instead of (4.14) we derive the estimate
Finally \(M_2\) is estimated by the same bound as \(M_1^1\).
Summarizing,
Using estimates (4.14), (4.15) and (4.16) in (4.10) yields
The last term on the r.h.s. of (4.8) can be bounded by
First, we estimate
Using the form of \(\xi _x\) yields
Exploiting the form of \(\xi _x\) and the imbedding
we obtain
Summarizing, we have
Expression \(A_2\) has the form
First, we estimate
Next, we estimate
Next \(A_2^3\le A_2^1\) and finally
Applying the Hölder inequality with respect to \(\xi \) yields
where \(1/p+1/p'=1\).
In view of imbeddings
which holds for \(3/2-3/2p'\le 1\), we finally obtain the estimate for \(\alpha \ge 1/2\)
Summarizing,
Since \(p_\xi \) in \(A_3\) plays the same role as \(v_{\xi \xi }\) in \(A_1\) we obtain from (4.19) the estimate
Using estimates (4.19), (4.20) and (4.21) in (4.18) implies
Exploiting estimates (4.9), (4.17) and (4.22) in (4.8), we obtain
where \(\alpha >5/8\).
Next, we estimate the norm
It is sufficient to estimate the last two term on the r.h.s. of (4.24) only. First we examine
Applying the Hölder inequality with respect to \(\xi '\), \(\xi ''\) in \(P_1\) yields
where \(1/p_1+1/p_2+1/p_3=1\), \(p_2<2\), \(\alpha '=\alpha +{1\over 2p_1}\big ({3\over 2}p_1-3\big )\).
we use imbeddings
and
From (4.25)–(4.28) and the form of \(\alpha '\) we obtain the restrictions
Eliminating \(p_1\), \(p_2\), \(p_3\) yields the restriction
Hence
Next, we examine \(P_2\),
where \(1/p_1+1/p_2=1\), \(p_2<2\), \(\alpha '=\alpha +{3\over 4}-{3\over 2p_1}\).
We use the imbeddings
Since restrictions (4.30), (4.31) can hold together we obtain the estimate
Finally,
where \(\lambda <8/3\) and \(\alpha >5/8\).
Using estimates (4.29), (4.32) and (4.33) in (4.25) implies the estimate
Finally, we examine
First, we estimate
Performing integration with respect to \(t''\) gives
The second term \(Q_2\) is bounded by
where \(\beta >\alpha \) and \(\beta \le 1+\alpha \).
Finally
where \(\beta >\alpha \), \(\beta \le 1/2+\alpha \).
Using estimates (4.36)–(4.38) in (4.35) yields
Using (4.34) and (4.39) in (4.24) we obtain the estimate
From (4.23) and (4.40) and recalling the index n we derive (4.4). This concludes the proof of Lemma 4.2. \(\square \)
From the properties of the space trace operators we have
Lemma 4.3
Let the assumptions of Lemma 4.2 hold. Then
In the next lemma we estimate norms of \(\nabla \varphi _n\) in the r.h.s. of (4.3).
Lemma 4.4
Assume that \({\bar{v}}_n\in V_2^{2+\alpha }({\mathop {\Omega }\limits ^{1}}_0^t)\), \(a>0\).
Then solutions to problem (4.1) satisfy
where \(g_n\) is defined in \((3.8)_2\).
Proof
In the proof we use the simplified notation
The aim of this proof is to find an estimate for the expression
where
where G is the Green function to the Dirichlet problem
Hence the Green function satisfies the condition
Moreover
In view of (4.45), (4.47), (4.48) and the Calderon–Zygmund estimate we get
Using definition (2.2.3) of the Besov space equivalent to definition (2.2.5) and applying the Calderon–Zygmund estimate, we obtain
Considering \(J_1\), we have
where
To estimate \(J_1^1\) we use the Hölder inequality with respect to \(\xi '\), \(\xi ''\) and the Minkowski inequality.
Then we obtain
where \(1/p+1/p'=1\), \(p'<2\), \(\alpha '=\alpha +{1\over 2p}\big ({3\over 2}p-3\big )\).
Next, we examine
where
Finally, \(J_2^2\) can be treated in the same way as \(J_1^1\). Then we obtain
Summarizing,
Finally, we examine \(I_3\). The Calderon–Zygmund theorem implies
where
Estimating \(L_1^1\) yields
Next, \(L_1^2\) is estimated by
Finally, (4.51) and (4.52) imply
Finally, we estimate \(L_2\),
First, we consider
By the Hölder inequality we get
where \(1/p+1/p'=1\), \(1/q+1/q'=1\), \(p'<2\), \(\alpha '={\alpha \over 2}+{1\over 2p}\big ({1\over 2}p-1\big )\).
Performing integration with respect to \(t''\) in the second factor yields
where we need the following restrictions
We see that the above inequalities do not imply any restriction.
Finally, we estimate
where \(1/p+1/p'=1\), \(1/q+1/q'=1\), \(p'<2\), \(\alpha '={\alpha \over 2}+{1\over 4}-{1\over 2p}\).
If the following restrictions hold
we obtain the estimate
Estimates (4.54) and (4.55) imply
From (4.53) and (4.56) we derive the bound
Using estimates (4.49), (4.50) and (4.57) in (4.44), we obtain
Recalling notation (4.43) in (4.58) gives (4.42). This ends the proof. \(\square \)
Corollary 4.5
Using estimates (4.4), (4.41) and (4.42) in (4.3) yields
5 Estimates for Solutions to Problem (3.9)
Lemma 5.1
Assume that \({\mathop {v}\limits ^{i}}_n\), \({\mathop {{\bar{H}}}\limits ^{i}}_n\in V_2^{2+\alpha }({\mathop {\Omega }\limits ^{i}}_0^t)\), \(i=1,2\) and \(a>0\). Assume also that \(H_{*i}\in H^{3/2+\alpha ,3/4+\alpha /2}(B^t)\), \(i=1,2\), \({\mathop {H}\limits ^{i}}(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{i}}_0)\), \(i=1,2\).
Then
Proof
Applying Lemma 2.5.1 to problem (3.9) yields the existence of solutions such that \({\mathop {{\bar{H}}}\limits ^{i}}_{n+1}\in V_2^{2+\alpha }({\mathop {\Omega }\limits ^{i}}_0^t)\) satisfying boundary conditions \((3.9)_{5,6,7}\) and initial conditions \((3.9)_8\). Moreover, Lemma 2.5.1 implies the estimate
where \({\mathop {{{\bar{\Phi }}}}\limits ^{i}}_n\) is a solution to (2.5.3) with \({\mathop {N}\limits ^{i}}_n\) which replaces \({\mathop {N}\limits ^{i}}\), \(i=1,2\).
Recalling the forms of quantities \({\mathop {M}\limits ^{i}}_n\), \({\mathop {N}\limits ^{i}}_n\), \(i=1,2\), and repeating the estimations appeared in the proofs of Lemmas 4.2 and 4.4, we obtain
and
Repeating the proof of (4.58) we obtain
Moreover, Sect. 7 implies the estimates
The above estimates imply (5.1). This concludes the proof. \(\square \)
A solution \({\mathop {{\bar{v}}}\limits ^{2}}\) to problem (3.2) appeared in problem (3.7) can be estimated by appropriate norms of \({\mathop {{\bar{v}}}\limits ^{1}}\).
Expressing (3.2) in Lagrangian coordinates we obtain
where \({\mathop {v}\limits ^{2}}(0)\) is a solution to problem (3.3)
We prove existence of solutions to problem (5.5) by the following method of successive approximations
Lemma 5.2
Let \({\mathop {{\bar{v}}}\limits ^{1}}\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^T)\), \({\mathop {v}\limits ^{2}}(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{2}}_0)\). Let \(\alpha >5/8\). Let \(D=c(\Vert {\mathop {{\bar{v}}}\limits ^{1}}\Vert _{H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{1}}_0^t)}+ \Vert {\mathop {v}\limits ^{2}}(0)\Vert _{H^{1+\alpha }({\mathop {\Omega }\limits ^{2}}_0)})\). For T sufficiently small there exists a solution to problem (5.5) such that \({\mathop {{\bar{v}}}\limits ^{2}}\in H^{2+\alpha ,1+\alpha /2}({\mathop {\Omega }\limits ^{2}}_0^t)\), \(t\le T\) and the estimate
where \(\gamma >1\) holds.
Proof
We use the method of successive approximations described by (5.7). Using the techniques from the proofs of Lemmas 4.2 and 4.4 we obtain
where \(a>0\).
Let \(\gamma >1\). Assume that
where
Then (5.9) yields
Let t be so small that
Then (5.11) yields
Assuming that \(X_0=0\) estimate (5.12) implies
Introducing differences
we can show convergence of the constructed sequence. This concludes the proof of Lemma 5.2. \(\square \)
In D introduced in assumptions of Lemma 5.2 we have the term \({\mathop {v}\limits ^{2}}(0)\) which is a solution to problem (5.6). Therefore, we need
Lemma 5.3
Assume that \({\mathop {v}\limits ^{1}}(0)\in H^{1/2+\alpha }(S_0)\). Then there exists a solution to (5.6) such that \({\mathop {v}\limits ^{2}}(0)\in H^{1+\alpha }({\mathop {\Omega }\limits ^{2}}_0)\) and the estimate holds
Corollary 5.4
Using (5.8) and (5.14) in (5.1) yields
6 Existence of Local Solutions to Problem (1.18), (1.20)
We prove existence of local solutions to problem (1.18), (1.20) by the method of successive approximations defined in Sect. 3.
Lemma 6.1
(uniform boundedness of the sequence introduced in Sect. 3) Assume that the quantity
is finite, where \(\alpha >5/8\).
Let
Then for t sufficiently small the following bound holds
where \(\gamma >1\) and \(c_0\) is the constant which estimates constants in (4.59) and (5.15).
Proof
Inequalities (4.59) and (5.15) imply
Let \(\gamma >1\). Assume that
Let t be so small that
Then (6.4) implies
Assume that \(X_0=0\). Then (6.5) and (6.7) imply (6.3). This ends the proof. \(\square \)
To show convergence of the above sequence we need
Lemma 6.2
Let the sequence \(\{{\bar{v}}_n,{\bar{p}}_n,{\mathop {{\bar{H}}}\limits ^{1}}_n,{\mathop {{\bar{H}}}\limits ^{2}}_n\}\) be examined in Lemma 6.1. Let
where
Then
where \(a>0\).
Proof
Taking the difference of (3.8) for n and for \(n-1\) we obtain
Taking the difference of (3.9) for n and for \(n-1\) yields
First we examine problem (6.10).
Let \(\Phi _n\) be a solution to the problem
There exists the Green function G(x, y) to problem (6.12) such that \(G(x,y)|_{S_0}=0\) and
Applying Lemma 2.3.2 with estimate (2.3.9) to problem (6.10) yields
We estimate a few terms from the r.h.s. of (6.14)
Continuing the considerations we derive
Applying Lemma 2.5.1 in the case \(\alpha =0\) to problem (6.11) yields
We shall estimate some terms from the r.h.s. of (6.16). Hence, we examine the term
where
and applying the Hölder inequality with respect to \(\xi \) yields
where \(1/p+1/p'=1\). Using the imbeddings
The above imbeddings can hold together because they imply the restriction
which holds for \(\alpha >1/2\). Then
Next, we estimate
Applying imbeddings (6.17) yields
Finally,
Summarizing and using Lemma 6.1 we obtain
Similar considerations can be applied to other terms from the r.h.s. of (6.16).
Estimating the terms from the r.h.s. of (6.16) we obtain
where \(a>0\). Estimates (6.15) and (6.18) imply (6.9). This ends the proof. \(\square \)
7 Problem (2.5.2) in a Neighborhood of \(S_0\)
In this Section we consider problem (2.5.2) localized to a neighborhood of any pooint of \(S_0\). The problem is complicated because it describes interaction of magnetic fields through the free surface \(S_0\). The localization means that problem (2.5.2) is multiplied by a function \(\zeta \) from the partition of unity with a support in a neighborhood of a point of \(S_0\). Then we obtain a problem for functions \({\mathop {{\bar{H}}}\limits ^{i}}\zeta \), \(i=1,2\). Next, we apply a transformation which makes \(S_0\) locally flat. Moreover, we consider a more general transmission condition. Therefore, we derive the problem
with transmission conditions
where \(a_i\), \(b_i\), \(i=1,2\), are some constants, \({{\bar{\tau }}}_1=(1,0,0)\), \({{\bar{\tau }}}_2=(0,1,0)\) and with the initial conditions
where \({\mathop {u}\limits ^{i}}(0)\), \(i=1,2\), are divergence free.
Since problem (7.1)–(7.3) is derived by the localization all data functions \(({\mathop {{\bar{f}}}\limits ^{i}},{\bar{k}}_i,{\bar{l}}_i,{\mathop {u}\limits ^{i}}(0),i=1,2)\) have compact supports.
In this Section we find estimates for solutions to problem (7.1)–(7.3) assuming that \({\mathop {{\bar{f}}}\limits ^{i}}\in H^{\alpha ,\alpha /2}({\mathbb {R}}_i^3\times (0,T))\), \({\mathbb {R}}_1^3=\{z:z_3>0\}\), \({\mathbb {R}}_2^3=\{z:z_3<0\}\), \({\bar{k}}_i\in H^{1+\alpha -1/2,1/2+\alpha /2-1/4}({\mathbb {R}}^2)\), \({\bar{l}}_i\in H^{2+\alpha -1/2,1+\alpha /2-1/4}({\mathbb {R}}^2)\), \({\mathbb {R}}^2=\{z:z_3=0\}\), \({\mathop {u}\limits ^{i}}(0)\in H^{1+\alpha }({\mathbb {R}}_i^3)\), \(i=1,2\), \(\alpha \in (0,1)\).
Remark 7.1
Functions \({\mathop {{\bar{f}}}\limits ^{i}}\), \(i=1,2\), are divergence free. This follows from the following construction of problem (7.1). Multiply equations \((2.5.1)_1\)–\((2.5.1)_4\) by cut-off functions \({\mathop {\zeta }\limits ^{1}}\) and \({\mathop {\zeta }\limits ^{2}}\), respectively. Introduce the notation \({\mathop {{\bar{v}}}\limits ^{i}}={\mathop {H}\limits ^{i}}{\mathop {\zeta }\limits ^{i}}\), \(i=1,2,\). Then \((2.5.1)_2\) and \((2.5.1)_4\) take the form
Introducing function \({\mathop {\varphi }\limits ^{i}}\) as a solution to the problem
we construct the function
which is divergence free and satisfies the equation
where
\(i=1,2\), \(\delta _{1i}\) is the Kronecker delta and \({\mathop {f}\limits ^{i}}\), \(i=1,2\), are divergence free. Passing to such coordinates that \(S_0\) becomes flat we derive system (7.1). Functions \({\mathop {\psi }\limits ^{i}}\) depend on \(\nabla {\mathop {H}\limits ^{i}}\), \({\mathop {H}\limits ^{i}}\), \(i=1,2\), so the dependence is not important in the proof of existence of solutions by the technique of regularizer. This ends Remark 7.1.
To make the initial data homogeneous we construct divergence free extensions \({\mathop {{\tilde{u}}}\limits ^{i}}\), \(i=1,2\), of initial data \({\mathop {u}\limits ^{i}}(0)\), \(i=1,2\), such that
Set
Then problem (7.1)–(7.3) takes the form
Expressing the transimission condition explicitly and extending (7.7) for \(t<0\) we have
Since \({\mathop {f}\limits ^{i}}\in H^{\alpha ,\alpha /2}({\mathbb {R}}_i^3\times {\mathbb {R}}_+)\), \(\alpha \in (0,1)\), we can extend them by zero on \({\mathbb {R}}^3\), respectively. We denote the extnsions by \({\mathop {f'}\limits ^{i}}\), \(i=1,2\). Then we are looking for solutions to the problems
Then the functions
are solutions to the problem with vanishing initial data
Lemma 7.2
Assume that \(({\mathop {w}\limits ^{1}},{\mathop {w}\limits ^{2}})\) is a solution to (7.11). Let \(\alpha \in (0,1)\). Assume that \(k_i\in H^{1/2+\alpha ,1/4+\alpha /2} ({\mathbb {R}}^2\times {\mathbb {R}}_+)\), \(i=1,2\), \(l_i\in H^{3/2+\alpha ,3/4+\alpha /2}({\mathbb {R}}^2\times {\mathbb {R}}_+)\), \(i=1,2\).
Then there exists a solution to problem (7.11) such that \({\mathop {w}\limits ^{i}}\in H^{2+\alpha ,1+\alpha /2}({\mathbb {R}}_i^3\times {\mathbb {R}}_+)\), \({\mathop {w}\limits ^{i}}|_{t=0}=0\), \(i=1,2\), and
Proof
Apply the Fourier-Laplace transform
\(\textrm{Re}\ s>0\), \(s=i\xi _0+\gamma \), \(\xi =(\xi _1,\xi _2)\), \(z'=(z_1,z_2)\), \(z'\cdot \xi =z_1\xi _1+z_2\xi _2\), \(z=(z_1,z_2,z_3)\), to problem (7.11). Then we have
where \(\tau _1^2=\mu _1\sigma _1s+\xi ^2\), \(\tau _2^2=\mu _2\sigma _2s+\xi ^2\).
Solving \((7.14)_{1,2}\) and using the Shapiro-Lopatinskii condition we obtain
Inserting \((7.15)_1\) in the transmission conditions \((7.14)_{3,4,5}\) yields
Using (7.15), we get
Using \((7.17)_3\) in \((7.17)_{1,2}\) and setting
we obtain
where
Solving (7.19) yields
We have the qualitative relations
where \({\tilde{h}}\), \(\tau \) replace \(({\tilde{h}}_1,{\tilde{h}}_2)\), \((\tau _1,\tau _2)\), respectively.
From (7.15) we have
Continuing,
where the summation over \(\alpha \in \{1,2\}\) is assumed. Using Lemmas 2.6.1 and 2.6.2 we conclude the proof of Lemma 7.2. \(\square \)
8 Initial-Boundary Value Problem Near B
In the proof of Lemma 2.5.1 we distinguish a local problem near B (see \(1_4\)). To examine the problem we localize (2.5.2) to a neighborhood of some point of B using an appropriate function from the partition of unity. Introducing a new system of coordinates with the origin at the point of B and flattening locally B we obtain
First we construct a function \({\mathop {v}\limits ^{1}}\) such that \(\textrm{div}\,{\mathop {v}\limits ^{1}}|_{x_3}=h\) so \({\mathop {v}\limits ^{1}}_{1,x_1}=h_1\), \({\mathop {v}\limits ^{1}}_{2,x_2}=h_2\), \({\mathop {v}\limits ^{1}}_{3,x_3}=h_3\), \(h=h_1+h_2+h_3\) on \(x_3=0\).
Next, we construct a function \({\mathop {v}\limits ^{2}}\) as a solution to the problem
Introducing the function
we see that it is a solution to the following initial-boundary value problem
Lemma 8.1
Let \({\mathbb {R}}^2=\{x\in {\mathbb {R}}^3:x_3=0\}\). Assume that \(h\in H^{{1\over 2}+\alpha ,{1\over 4}+{\alpha \over 2}}({\mathbb {R}}^2\times {\mathbb {R}}_+)\). Then \({\mathop {v}\limits ^{1}}\in H^{2+\alpha ,1+\alpha /2}({\mathbb {R}}_+^3\times {\mathbb {R}}_+)\) and
Assume that \(g\in H^{\alpha ,\alpha /2}({\mathbb {R}}_+^3\times {\mathbb {R}}_+)\). Then there exists a solution to (8.2) such that \({\mathop {v}\limits ^{2}}\in H^{2+\alpha ,1+\alpha /2}({\mathbb {R}}_+^3\times {\mathbb {R}}_+)\) and
Lemma 8.2
Assume that \(a_j\in H^{3/2+\alpha ,3/4+\alpha /2}({\mathbb {R}}^2\times {\mathbb {R}}_+)\), \(j=1,2\), \(\alpha \in (0,1)\). Then there exists a solution to problem (8.4) such that \(w\in H^{2+\alpha ,1+\alpha /2}({\mathbb {R}}_+^3\times {\mathbb {R}}_+)\), w is divergence free and
Proof
Applying the Fourier–Laplace transform (7.13) to (8.4) yields
where the summation convention over repeated Greek indices from 1 to 2 is assumed.
Simplifying (8.8) yields
Multiplying \((8.9)_1\) by \(\xi _\beta \) and summing with respect to \(\beta \) gives
Introducing the quantity
we obtain from (8.10), \((8.9)_{1,2}\) the following problem:
From \((8.12)_2\) we have
Inserting this in \((8.12)_1\) yields
Since Re \(s>0\) we get
Solving (8.15) gives
Since \(\textrm{Re}\sqrt{s+\xi ^2}>0\), we have to assume that \(c_1=0\). From boundary condition \((8.15)_2\) we obtain \(c_2=\xi _\beta {\tilde{a}}_\beta \). Hence
In view of (8.11) we have
so
Hence
In view of (8.13) and (8.17) we have
Differentiating (8.19) with respect to \(z_3\) and projecting on the plane \(z_3=0\) we obtain
Applying the inverse Laplace–Fourier transform we obtain condition \((8.4)_3\).
Finally, applying Lemmas 2.6.1 and 2.6.2 we derive (8.7). This ends the proof. \(\square \)
Data Availability Statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Kacprzyk, P., Zaja̧czkowski, W.M.: On the Faedo–Galerkin method for a free boundary problem for incompressible viscous magnetohydrodynamics. Topol. Methods Nonlinear Anal. 52, 69–98 (2018)
Shibata, Y., Zaja̧czkowski, W.M.: On local solutions to a free boundary problem for incompressible viscous magnetohydrodynamics in the \(L_p\)-approach. Diss. Math. 566, 1–102 (2021)
Cole, G.H.A.: Fluid Dynamics. London (1962)
Landau, L.D., Lifshitz, E.M., Pitaevskii, L.P.: Electrodynamics of Continuous Media. Landau and Lifshitz Course of Theoretical Physics, vol. 8, 2nd edn. Elsevier, Amsterdam (2013)
Ladyzhenskaya, O.A., Solonnikov, V.A.: Solvability of some nonstationary problems of magnetohydrodynamics for viscous incompressible fluids. Trudy Mat. Inst. Steklova 59, 115–173 (1960). ((in Russian))
Padula, M., Solonnikov, V.A.: On free boundary problem of mhd. Zap. Nauchn Sem. POMI 385, 135–186 (2010). ((in Russian); English transl. J. Math. Sc. (N.Y.) 178, 313–344 (2011))
Frolova, E.V.: Free boundary problem of magnetohydrodynamics. Zap. Nauchn. Sem. POMI 425, 149–178 (2014). ((in Russian))
Frolova, E.V., Solonnikov, V.A.: Solvability of a free boundary problem of magnetohydrodynamics in an infinite time interval. Zap. Nauchn. Sem. POMI 410, 131–167 (2013). ((in Russian), English transl. J. Math. Sci. (N.Y.)195 (2013), 76–97)
Kacprzyk, P.: Local free boundary problem for incompressible magnetohydrodynamics. Diss. Math. 509, 1–52 (2015)
Kacprzyk, P.: Local existence of solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid. Appl. Math. 30(4), 461–488 (2003)
Zaja̧czkowski, W.M.: On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface. Diss. Math. 324, 101 (1993)
Frolova, E.V., Shibata, Y.: On the maximal \(L_p-L_q\) regularity theorem for the linearized electromagnetic field equations with interface condition. Zap. Nauchn. Sem. POMI 489, 130–172 (2020). (English transl.: J. Math. Sci. 260 (2022), 87–117)
Frolova, E.V., Shibata, Y.: Local well posedness for the magnetohydrodynamics in the different two liquids case. Mathematics 10, 4751 (2022)
Frolova, E.V.: Linearization of a free boundary problem of magnetohydrodynamics. Probl. Math. Anal. 95, 69–78 (2018). (English transl.: J. Math. Sci. 235 (2018), 322–333)
Kacprzyk, P.: Global free boundary problem for incompressible magnetohydrodynamics. Diss. Math. 510, 1–44 (2015)
Kacprzyk, P.: Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid. Appl. Math. 31(1), 69–77 (2004)
Kacprzyk, P.: Free boundary problem for the equations of magnetohydrodynamic incompressible viscous fluid. Appl. Math. (Warsaw) 34, 75–95 (2007)
Besov, O.V., Il’in, V.P., Nikolskii, S.M.: Integral Representations of Functions and Imbedding Theorems. Nauka, Moscow (1975). ((in Russian); English transl.: Scripta Series in Mathematics, Winston and Halsted Press, 1979)
Golovkin, K.K.: On equivalent norms for fractional spaces. Trudy Mat. Inst. Stekolva 66, 364–383 (1962). ((in Russian))
Bugrov, Y.S.: Function spaces with mixed norm. Izv. AN SSSR Ser. Mat. 35, 1137–1158 (1971). (English transl.: Math. USSR-Izv., 5, 1145–1167 (1971))
Rencławowicz, J., Zaja̧czkowski, W.M.: On the Stokes system in cylindrical domains. J. Math. Fluid Mech. 24, 64 (2022). https://doi.org/10.1007/s00021-022-00698-z
Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)
Solonnikov, V.A.: Estimates of solutions to nonstationary linearized Navier–Stokes system. Trudy Mat. Inst. Steklova 70, 213–317 (1964). ((in Russian))
Solonnikov, V.A.: On an unsteady motion of an isolated volume of a visous incompressible fluid. Izv. Akad. Nauk SSSR Ser. Mat. 51, 1065–1087 (1987). ((in Russian))
Solonnikov, V.A.: An initial boundary value problem for Stokes system that arises in the study of a problem with a free boundary. Trudy Mat. Inst. Steklova 188, 150–188 (1990). ((Russian); English transl.: Proc. Steklov Inst. Math. 3 (1991), 191–239)
Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow (1967). ((in Russian))
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by Y. Shibata.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kacprzyk, P., Zaja̧czkowski, W.M. Existence of Local Solutions to a Free Boundary Problem for Incompressible Viscous Magnetohydrodynamics. J. Math. Fluid Mech. 26, 50 (2024). https://doi.org/10.1007/s00021-024-00879-y
Accepted:
Published:
DOI: https://doi.org/10.1007/s00021-024-00879-y
Keywords
- Free boundary
- Incompressible magnetohydrodynamics with resistivity
- Regularizer technique
- Transmission conditions on the free surface
- The method of successive approximations