Abstract
This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal \(L_p\)–\(L_q\) regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of \({\mathcal R}\)-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the \({{\mathcal {R}}}\)-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; \({{\mathcal {R}}}\) boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (\({{\mathcal {R}}}\)-solvers and their application to periodic \(L_p\) estimates, Preprint in 2019) for the \(L_p\) boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.
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1 Introduction
This paper is concerned with time-periodic solutions of one-phase and two-phase problems for the Navier–Stokes equations. The periodic solutions for the Navier–Stokes equations have been studied in many articles [3,4,5,6,7,8, 10,11,12,13,14, 20, 23] and references therein. One well-known approach to prove the existence of periodic solutions is the utilization of the Poincaré operator, which maps an initial value into the solution of the PDE at time \({\mathcal T}\), where \({\mathcal T}\) is the period of the data. A fixed point of the Poincaré operator yields an initial value that induces a \({\mathcal T}\)-time-periodic solution. Such a utilization of the Poincaré operator is naturally carried out under the global well-posedness of the corresponding initial-boundary value problem for the bounded data on the right hand side of the equations. In the bounded domain case, this is deeply related with the situation where 0 does not belong to the spectrum of the system of the linearized equations. However, in many interesting problems in mathematical physics, we meet the situation that 0 is in the spectrum. One-phase or two-phase problems for the Navier–Stokes equations are typical examples. As explained in Sects. 1 and 2, the one-phase and two-phase problems we treat in this paper are formulated by the Navier–Stokes equations with free boundary condition or transmission condition on the interface in a time-dependent domain \(\Omega _t\), which is also unknown. Usually, \(\Omega _t\) is transformed to a fixed domain \(\Omega \) by introducing an unknown function representing the boundary or the interface of \(\Omega _t\). Thus, the problem treated here becomes a quasilinear system of equations with nonlinear boundary or transmission conditions. The first of our key approaches is to separate solutions into stationary part and oscillatory part. Then, the zero eigen-value of the linearized equations appears only in the equations for the stationary problem. We change the linearized equations by using some necessary conditions for the unique existence of solutions to avoid eigen-value 0 for the linearized problem. This technique is possible under the separation of the stationary part and the oscillatory part, which does not appear when working with the Poincaré operator. The second is to introduce a systematic approach to the maximal \(L_p\)–\(L_q\) regularity for the oscillatory part based solely on the \({\mathcal R}\)-solver for the resolvent problem of the linearized equations developed in [15,16,17,18,19] and a transference theorem for the \(L_p\) boundedness of the operator-valued Fourier multiplier due to Eiter, Kyed and Shibata in [2]. The \(L_p\)–\(L_q\) maximal regularity for the oscillatory part of solutions is necessary because our problem is a quasilinear system with non-homogeneous boundary conditions. Since the maximal regularity for the oscillatory part of the periodic solutions does not seem to be well-studied, our systematic approach gives a quite important contribution to the study of systems of parabolic equations with non-homogeneous boundary conditions, which is the novelty of this paper.
1.1 One-phase problem
Let \(\Omega _t\) be a time-dependent domain in the N-dimensional Euclidean space \({\mathbb R}^N\) (\(N \ge 2\)). Let \(\Gamma _t\) be the boundary of \(\Omega _t\) and \(\mathbf{n}_t\) the unit outer normal to \(\Gamma _t\). We assume that \(\Omega _t\) is occupied by some incompressible viscous fluid of unit mass density whose viscosity coefficient is a positive constant \(\mu \). Let \(\mathbf{u}= {}^\top (u_1(x, t), \ldots , u_N(x,t))\), \(x = (x_1, \ldots , x_N)\in \Omega _t\), and \({\mathfrak {p}}={\mathfrak {p}}(x, t)\) be the velocity field and the pressure field in \(\Omega _t\), respectively, where \({}^\top M\) denotes the transposed of M. We consider the Navier–Stokes equations in \(\Omega _t\) with free boundary condition as follows:
for \(t \in {\mathbb R}\). Here, \(\mathbf{f}= \mathbf{f}(x, t)\) is a prescribed time-periodic external force with period \(2\pi \); \(H(\Gamma _t)\) denotes the \((N-1)\)-fold mean curvature of \(\Gamma _t\) which is given by \(H(\Gamma _t)\mathbf{n}_t = \Delta _{\Gamma _t}x\) for \(x \in \Gamma _t\), where \(\Delta _{\Gamma _t}\) is the Laplace–Beltrami operator on \(\Gamma _t\); \(V_{\Gamma _t}\) is the evolution speed of \(\Gamma _t\) along \(\mathbf{n}_t\); \(\sigma \) is a positive constant representing the surface tension coefficient; \(\mathbf{D}(\mathbf{u})\) is the doubled deformation tensor given by \(\mathbf{D}(\mathbf{u}) = \nabla \mathbf{u}+ {}^\top \nabla \mathbf{u}\); and \(\mathbf{I}\) is the \((N\times N)\)-identity matrix. Moreover, for any \((N\times N)\)-matrix of functions \(\mathbf{K}\) whose \((i, j)\mathrm{th}\) component is \(K_{ij}\), \(\mathrm{Div}\,K\) is an N-vector whose \(i\mathrm{th}\) component is \(\sum _{j=1}^n\partial _jK_{ij}\) and for any N-vector of functions \(\mathbf{v}= {}^\top (v_1, \ldots , v_N)\), \(\mathbf{v}\cdot \nabla \mathbf{v}\) is an N-vector of functions whose \(i\mathrm{th}\) component is \(\sum _{j=1}^Nv_j\partial _j v_i\), where \(\partial _j=\partial /\partial x_j\).
Our problem is to find \(\Omega _t\), \(\Gamma _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfying the periodic condition:
for any \(t \in {\mathbb R}\).
To state the main result, we introduce assumptions and some functional spaces. Let \(\mathbf{p}_i = \mathbf{e}_i ={}^T(0, \ldots , 0, \overset{\mathrm{i-th}}{1}, 0, \ldots , 0)\) for \(i=1, \ldots , N\) and \(\mathbf{p}_\ell \) (\(\ell =N+1, \ldots , M)\) be one of \(x_i\mathbf{e}_j - x_j\mathbf{e}_i\) (\(1 \le i< j \le N\)). Notice that \(\mathbf{p}_\ell \) forms a basis of the rigid space \(\{ \mathbf{v}\mid \mathbf{D}(\mathbf{v}) = 0\}\) and the number M is its dimension. We will construct \(\Omega _t\) satisfying the following two conditions:
Here and in the following, \((M_{\ell ,m})_{\ell , m=1,\ldots , N}\) denotes an \((N\times N)\)-matrix whose \((\ell , m)\mathrm{th}\) component is \(M_{\ell , m}\); for any domain G and \((N-1)\)-dimensional hypersurface S, we let
where \(\overline{g(x)}\) denotes the complex conjugate of g(x), and \(\mathrm{d}\sigma \) the surface element of S. |G| denotes the Lebesgue measure of a Lebesgue measurable set G of \({\mathbb R}^N\); and \(B_R\) is the ball with radius R centered at the origin. For \(1< p < \infty \) and any Banach space X with norm \(\Vert \cdot \Vert _X\), let
where \(\dot{f}\) denotes the derivative of f with respect to t. Let
For any domain G in \({\mathbb R}^N\) and \(1 \le q \le \infty \), \(L_q(G)\), \(H^m_q(G)\), and \(B^s_{q,p}(G)\) denote the standard Lebesgue, Sobolev, and Besov spaces on G, and \(\Vert \cdot \Vert _{L_q(G)}\), \(\Vert \cdot \Vert _{H^m_q(G)}\), and \(\Vert \cdot \Vert _{B^s_{q,p}(G)}\) denote their respective norms. For any integer d, \(X^d\) denotes the d-fold product of the space X, that is \(X^d = \{\mathbf{g}= {}^\top (g_1, \ldots , g_d) \mid g_j \in X (j=1, \ldots , d)\}\), while the norm of \(X^d\) is denoted by \(\Vert \cdot \Vert _X\) instead of \(\Vert \cdot \Vert _{X^d}\) for simplicity.
The following theorem is our main result concerning time-periodic solutions of the one-phase problem for the Navier–Stokes equations.
Theorem 1
Let \(1< p, q < \infty \) and \(2/p + N/q < 1\). Let \(D\subset B_R\) be a domain. Then, there exists a positive constant \(\epsilon \) and an injective map \(x = \Phi (y, t) : B_R \rightarrow {\mathbb R}^N\) for each \(t \in (0, 2\pi )\) with
for which the following assertion holds: If \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(D)^N)\) satisfies the support condition: \(\mathrm{supp}\,\mathbf{f}(\cdot , t) \subset D\) for any \(t \in (0, 2\pi )\), the orthogonal condition
and the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D)^N)} \le \epsilon \), then there exist \(\mathbf{v}(y, t)\), \({\mathfrak {q}}(y, t)\), and \(\rho (y, t)\) with
such that
where \(\Phi ^{-1}(x, t)\) is the inverse map of the correspondence: \(x = \Phi (y, t)\) for any \(t \in (0, 2\pi )\), are solutions of equations (1.2) satisfying the periodicity condition (1.2), and \(\Gamma _t\) is given by
where \(\xi (t)\) is the barycenter point of \(\Omega _t\) defined by setting
Moreover, \(\mathbf{v}\) and \(\rho \) satisfy the estimate:
for some constant C independent of \(\epsilon \).
Remark 2
In the construction of the map \(\Phi \), we see that \(\Phi (y,t) = y + R^{-1}\rho (y, t) + \xi (t)\) for \(y \in S_R\).
1.2 Two-phase problem
Let \(\Omega _{+t}\) be a time-dependent domain in the N-dimensional Euclidean space \({\mathbb R}^N\). Let \(\Gamma _t\) be the boundary of \(\Gamma _t\) and \(\mathbf{n}_t\) its unit outer normal. Let \(\Omega \) be a bounded domain in \({\mathbb R}^N\) and S the boundary of \(\Omega \). We assume that \(\Omega _{+t} \subset \Omega \) and \(\Gamma _t \cap S = \emptyset \). Let \(\Omega _{-t} = \Omega {\setminus }(\Omega _{+t} \cup \Gamma _t)\) and set \(\Omega _t = \Omega _{+t} \cup \Omega _{-t}\). We assume that \(\Omega _{\pm t}\) be occupied by some incompressible viscous fluids of unit mass densities whose viscosity coefficients are positive constants \(\mu _\pm \). Let \(\mathbf{u}= {}^\top (u_1, \ldots , u_N)\) and \({\mathfrak {p}}\) be the velocity field and the pressure field on \(\Omega _t\), respectively. We consider the following Navier–Stokes equations with transmission condition on \(\Gamma _t\) and no-slip condition on S:
for \(t\in {\mathbb R}\), where \(\mathbf{f}=\mathbf{f}(x, t)\) is a prescribed time-periodic external force with period \(2\pi \); \(\mu \) is the viscosity coefficient given by
and [[f]] denotes the jump of \(f_\pm \) defined on \(\Omega _\pm \) along \(\mathbf{n}_t\) defined by setting
The purpose of this paper is also to find \(\Omega _{\pm t}\), \(\Gamma _t\), \(\mathbf{u}_\pm \) and \({\mathfrak {p}}_\pm \) which satisfy the periodicity condition:
To state a main result, we introduce the assumptions about \(\Omega _t\) as follows. We assume that \(\Omega \supset B_R\) for some \(R > 0\), and that
The following theorem is our main result concerning time-periodic solutions of the two-phase problem for the Navier–Stokes equations.
Theorem 3
Let \(1< p, q < \infty \) and \(2/p + N/q < 1\). \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\). Then, there exist a positive constant \(\epsilon \) and a bijective map \(x = \Phi (y, t)\) from \(\Omega \) onto itself such that for any \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N)\) satisfying the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(\Omega ))} \le \epsilon \), there exist \(\mathbf{v}_\pm (y, t)\), \({\mathfrak {q}}_\pm (y, t)\) and \(\rho (y, t)\) with
such that
where \(y = \Phi ^{-1}(x, y)\) is the inverse map of \(x = \Phi (y, t)\), are solutions of problem (1.9), and \(\Gamma _t\) is given by
where \(\xi (t)\) is the barycenter point of \(\Omega _+\) defined by setting
Moreover, \(\mathbf{v}_\pm \) and \(\rho \) satisfy the estimate:
for some constant C independent of \(\epsilon \).
Method Since the domain \(\Omega _t\) is unknown, using the Hanzawa transform, we reduce the equations onto a fixed domain, which results in a system of quasilinear equations. Thus, we cannot use the analytic \(C_0\)-semi-group approach. Our main tool is to use the \(L_p\)-\(L_q\) maximal regularity for periodic solutions to the linearized equations, which can be obtained by using the \({\mathcal R}\)-solver to the generalized resolvent problem and applying the transference theorem ([1, 2]) to the solution formula represented by the \({\mathcal R}\)-solver. This is a quite new and more direct approach and a completely different idea than exploiting the Poincaré operator.
Further notation This section is ended by explaining further notation used in this paper. We denote the sets of all complex numbers, real numbers, integers, and natural numbers by \({\mathbb C}\), \({\mathbb R}\), \({\mathbb Z}\), and \({\mathbb N}\), respectively. Let \({\mathbb N}_0 = {\mathbb N}\cup \{0\}\). Let X be a Banach space with norm \(\Vert \cdot \Vert _X\). For any X-valued function \(f:{\mathbb R}\rightarrow X\) the functions \({\mathcal F}[f]\) and \({\mathcal F}^{-1}[f]\) denote the Fourier transform and the inverse Fourier transform of f, respectively, defined by setting
Let \(g:{\mathbb T}\rightarrow X\) be an X-valued function defined on the torus \({\mathbb T}= {\mathbb R}/ 2\pi {\mathbb Z}\). We define the Fourier transform \({\mathcal F}_{\mathbb T}\) acting on g by setting
which is regarded as a correspondence \(g \mapsto ({\mathcal F}_{\mathbb T}[g](k)) = \{{\mathcal F}_{\mathbb T}[g](k) \in X \mid k \in {\mathbb Z}\}\). For any sequence \((a_k) = \{a_k \in X \mid k \in {\mathbb Z}\}\), we define the inverse Fourier transform \({\mathcal F}^{-1}_{\mathbb T}\) acting on \((a_k)\) by setting
For any X-valued periodic function f with period \(2\pi \), we set
The \(f_S\) and \(f_\perp \) are called stationary part and oscillatory part of f, respectively.
For \(1 \le p \le \infty \), \(L_p({\mathbb R}, X)\) and \(H^1_p({\mathbb R}, X)\) denote the standard Lebesgue and Sobolev spaces of X-valued functions defined on \({\mathbb R}\), and \(\Vert \cdot \Vert _{L_p({\mathbb R}, X)}\), \(\Vert \cdot \Vert _{H^1_p({\mathbb R}, X)}\) denote their respective norms. For \(\theta \in (0, 1)\), \(H^\theta _{p, \mathrm{per}}((0,2\pi ), X)\) denotes the X-valued Bessel potential space of periodic functions defined by
As usual, we set \(L_{p, \mathrm{per}}((0, 2\pi ), X)=H^0_{p, \mathrm{per}}((0, 2\pi ), X)\).
For any multi-index \(\alpha = (\alpha _1, \ldots , \alpha _N) \in {\mathbb N}_0^N\) we set \(\partial _x^\alpha h = \partial _1^{\alpha _1}\cdots \partial _N^{\alpha _N} h\) with \(\partial _i = \partial /\partial x_i\). For any scalar function f, we write
where \(\partial _x^0f = f\). For any m-vector of functions \(\mathbf{f}={}^\top (f_1, \ldots , f_m)\), we write
For any N-vector of functions, \(\mathbf{u}={}^\top (u_1, \ldots , u_N)\), sometimes \(\nabla \mathbf{u}\) is regarded as an \((N\times N)\)-matrix of functions whose \((i, j)\mathrm{th}\) component is \(\partial _ju_i\). For any m-vector \(V=(v_1, \ldots , v_m)\) and n-vector \(W=(w_1, \ldots , w_n)\), \(V\otimes W\) denotes an \((m\times n)\) matrix whose \((i, j)\mathrm{th}\) component is \(V_iW_j\). For any \((mn\times N)\)-matrix \(A=(A_{ij, k} \mid i=1, \ldots , m, j=1, \ldots , n, k=1, \ldots , N)\), \(AV\otimes W\) denotes an N-column vector whose \(k\mathrm{th}\) component is the quantity: \(\sum _{j=1}^m\sum _{j=1}^n A_{ij, k}v_iw_j\).
Let \(\mathbf{a}\cdot \mathbf{b}=<\mathbf{a}, \mathbf{b}>= \sum _{j=1}^Na_jb_j\) for any N-vectors \(\mathbf{a}=(a_1, \ldots , a_N)\) and \(\mathbf{b}=(b_1, \ldots , b_N)\). For any N-vector \(\mathbf{a}\), let \(\Pi _0\mathbf{a}= \mathbf{a}_\tau : = \mathbf{a}- <\mathbf{a}, \mathbf{n}>\mathbf{n}\). For any two \((N\times N)\)-matrices \(\mathbf{A}=(A_{ij})\) and \(\mathbf{B}=(B_{ij})\), the quantity \(\mathbf{A}:\mathbf{B}\) is defined by \(\mathbf{A}:\mathbf{B}= \sum _{i,j=1}^NA_{ij}B_{ji}\). For any domain G with boundary \(\partial G\), we set
where \(\overline{\mathbf{v}(x)}\) is the complex conjugate of \(\mathbf{v}(x)\) and \(\mathrm{d}\sigma \) denotes the surface element of \(\partial G\). Given \(1< q < \infty \), let \(q' = q/(q-1)\). For \(L > 0\), let \(B_L = \{x \in {\mathbb R}^N \mid |x| < L\}\) and \(S_L = \{x \in {\mathbb R}^N \mid |x| = L \}\).
For two Banach spaces X and Y, \(X+Y = \{x + y \mid x \in X, y\in Y\}\), \({\mathcal L}(X, Y)\) denotes the set of all bounded linear operators from X into Y and \({\mathcal L}(X, X)\) is written simply as \({\mathcal L}(X)\). Moreover, let \({\mathcal R}_{{\mathcal L}(X, Y)}(\{{\mathcal T}(\lambda ) \mid \lambda \in I\})\) be the \({\mathcal R}\)-bound of the operator family \(\{{\mathcal T}(\lambda ) \mid \lambda \in I\}\subset {\mathcal L}(X, Y)\) (see also Definition 7). Let
The letter C denotes a generic constant and \(C_{a,b,c,\ldots }\) denotes that the constant \(C_{a,b,c,\ldots }\) depends on a, b, \(c, \ldots \); the value of C and \(C_{a,b,c,\ldots }\) may change from line to line.
2 Linearization principle
We now formulate the problems (1.1) and (1.9) in a fixed domain and state main results in this setting. Theorems 1 and 3 follow from the main theorems of this section.
2.1 One-phase problem
Let \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfies equations (1.1) and the periodicity condition (1.2). We have
Multiplying the first equation in (1.1) with \(\mathbf{p}_\ell \) and integrating the resultant formula on \(\Omega _t\) and using the divergence theorem of Gauss give that
In fact, we have used the fact that
which follows from the Reynolds transport theoremFootnote 1 and that \(\mathrm{div}\,\mathbf{u}=0\) in \(\Omega _t\). Thus, the periodicity condition (1.2) yields that
where we have used the assumption that \(\mathrm{supp}\, \mathbf{f}(\cdot , t) \subset D\) for any \(t \in {\mathbb R}\). Thus, the condition (1.6) is a necessary one to prove Theorem 1. From this observation, instead of problem (1.2), we consider the following equations:
for \(t \in {\mathbb R}\). In fact, if \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (2.2), then we have
which, combined with the periodicity condition (1.2), the assumption (1.3) and (2.1), leads to
Thus, \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy the first equation in (1.1). Therefore, under the stated assumptions, a solution to problem (2.2) is a solution to the original problem (1.1). However, as we shall see below, the condition (2.1) is not necessary to find a solution to (2.2).
From now on, we consider problem (2.2). We reduce problem (2.2) to some nonlinear equations on \(B_R\) by using the Hanzawa transform, which we explain below. Let \(\xi (t)\) be the barycenter point of \(\Omega _t\) defined by setting
where we have used the fact that \(|\Omega _t| = |B_R|\), which follows from the assumption (1.5). By the Reynolds transport theorem, we see that
because \(\mathrm{div}\,\mathbf{u}=0\). Let \(\rho (y, t)\) be an unknown time-periodic function with period \(2\pi \) such that
where \(S_R = \{x \in {\mathbb R}^N \mid |x| = R\}\) and \(\mathbf{n}\) is the unit outer normal to \(S_R\), that is \(\mathbf{n}= x/|x|\) for \(x \in S_R\). Let \(H_\rho \) be a suitable extension of \(\rho \) to \({\mathbb R}^N\), and then by the K-method in the theory of real interpolation [9, 21], we see that there exist constants \(C_1\) and \(C_2\) such that
for any \(t \in (0, 2\pi )\). In the following, we fix the method of this extension. For example, \(\hat{H}_\rho \) is the unique solution of the Dirichlet problem:
Let \(\varphi \) be a \(C^\infty ({\mathbb R}^N)\) function which equals one for \(x \in B_{2R}\) and zero for \(x \not \in B_{3R}\), and we set \(H_\rho = \varphi \hat{H}_\rho \). We assume that
with some small constant \(\delta > 0\). Notice that \(y/|y| = R^{-1}y\) for \(y \in S_R\) is the unit outer normal to \(S_R\). Let \(\Phi (y, t) = y + R^{-1}H_\rho (y, t)y+ \xi (t)\). We choose \(\delta > 0\) so small that the map \(x = \Phi (y,t)\) is injective. In fact, for any \(y_1\) and \(y_2\)
which leads to the injectivity of the transformation \(x = \Phi (y, t)\) for any \(t \in {\mathbb R}\) provided that \(0< \delta < 1\). Moreover, using the inverse mapping theorem, we see that the map \(x=\Phi (y, t)\) is surjective from \({\mathbb R}^N\) onto \({\mathbb R}^N\).
Let
Let \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy equations (1.1), and let \(\mathbf{v}(y, t) = \mathbf{u}(x, t)\) and \({\mathfrak {q}}(y, t) = {\mathfrak {p}}(x, t)\). We derive an equation for \(\mathbf{v}\) and \(\rho \) from the kinematic condition: \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) on \(\Gamma _t\). From the definition:
To represent \(\xi '(t)\), we introduce the Jacobian J(t) of the transformation \(x = \Phi (y, t)\), which is written as \(J(t) = 1 + J_0(t)\) with
Choosing \(\delta > 0\) small enough in (2.6), we have
From (2.4) it follows that
and so noting that \(\mathbf{n}\cdot \mathbf{n}=1\), we have the kinematic equation:
with
As will be seen in Sect. 3, we have \(<H(\Gamma _t)\mathbf{n}_t, \mathbf{n}_t> = (\Delta _{S_R} + (N-1)/R^2)\rho -(N-1)/R +\) nonlinear terms, and \(-(N-1)/R^2\) is the first eigen-value of the Laplace-Beltrami operator \(\Delta _{S_R}\) on \(S_R\) with eigen-functions \(y_j/R\) for \(y=(y_1, \ldots , y_N) \in S_R\). We need to derive some auxiliary equations to avoid the zero and first eigen-values of \(\Delta _{S_R}\). From the assumption (1.5) and the representation formulas of \(\Omega _t\) and \(\Gamma _t\) in (2.7), by using polar coordinates we have
and so we have
where \(\mathrm{d}\omega \) denotes the surface element of \(S_R\). Moreover, from (2.3) and the assumption (1.5), using polar coordinates centered at \(\xi (t)\), we have
from which it follows that
for \(j=1, \ldots , N\). Thus, under the assumption (1.5) and the representation of \(\Gamma _t\) and \(\Omega _t\) in (2.7), the kinematic condition (2.10) is equivalent to the equation
with
Therefore, to prove the existence of \((\Omega _t, \mathbf{u}, {\mathfrak {p}})\), we shall prove the well-posedness of the following equations:
where we have set
where \(\Delta _{S_1}\) is the Laplace–Beltrami operator on the unit sphere \(S_1\). For the functions on the right side of equations (2.16), \(\mathbf{G}(y, t)\) and \(\mathbf{F}(\mathbf{v}, \rho )\) are given in (3.13) in Sect. 3, \(g(\mathbf{v}, \rho )\) and \(\mathbf{g}(\mathbf{v}, \rho )\) given in (3.6) in Sect. 3, \(\tilde{d}(\mathbf{v}, \rho )\) has been given in (2.15) and \(\mathbf{h}(\mathbf{v}, \rho ) = (\mathbf{h}'(\mathbf{v}, \rho ), h_N(\mathbf{v}, \rho ))\) is given in (3.31) and (3.34) in Sect. 3.
The following theorem is the unique existence theorem of \(2\pi \)-periodic solutions of problem (2.16).
Theorem 4
Let \(1< p, q < \infty \) and \(2/p+N/q < 1\). Then, there exists a small constant \(\epsilon > 0\) such that if \(\mathbf{f}\) satisfies the assumption (1.6) and the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} \le \epsilon \), then problem (2.16) admits \(2\pi \)-periodic solutions \(\mathbf{v}\), \({\mathfrak {q}}\), and \(\rho \) satisfying the regularity condition (1.7) and the estimate (1.8) in Theorem 1.
Proof of Theorem 1
We prove Theorem 1 with the help of Theorem 4. Let \(\xi (t)\) be defined by
where c is chosen in such a way that
Here, \(\xi '(t)\) is given by the formula in (2.9). Then, we define \(\Omega _t\) and \(\Gamma _t\) by the formulas in (2.7). Let \(\Phi (y, t)=y+ R^{-1}H_\rho y+ \xi (t)\). By choosing \(\epsilon \) sufficiently small, estimates (1.8) and (2.5) ensure that the condition (2.6) is satisfied with small \(\delta > 0\). This yields the existence of the inverse map \(y= \Phi ^{-1}(x, t)\) of the map: \(x = \Phi (y, t)\). Thus, the velocity field \(\mathbf{u}(x, t)\) and the pressure \({\mathfrak {p}}(x, t)\) on \(\Omega _t\) are well-defined by setting \(\mathbf{u}(x, t) = \mathbf{v}(y, t)\) and \({\mathfrak {p}}(x, t) = {\mathfrak {q}}(y,t)\). Since \(\mathrm{div}\,\mathbf{u}=0\) in \(\Omega _t\), \(|\Omega _t|\) is a constant, and so \(|\Omega _t| = |B_R|\) by assumption (1.5). Moreover, if we set
then
and so \(\eta (t) = \xi (t) + d\) with some constant d. We assume that the assumption (1.4) holds, and then by (2.18) we have
which leads to \(d=0\), that is
Combining this with (1.5) gives that
which yields that \(\rho \) satisfies the equation:
Therefore, the kinematic equation: \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) holds on \(\Gamma _t\). So far, we see that \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (2.2). Since \(D \subset B_R\), there exists a constant \(\epsilon _0 > 0\) for which \(D \subset B_{R-3\epsilon _0}\). Since \(\Omega _t\) is a small perturbation of \(B_R\), choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(B_{R-\epsilon _0} \subset \Omega _t\), and so by (1.6) we have
Multiplying the first equation in (2.2) with \(\mathbf{p}_\ell \), integrating the resultant formulas with respect to x on \(\Omega _t\) and with respect to t on \((0, 2\pi )\), and using the periodicity (1.2) and (2.19) we have
for \(\ell =1, \ldots , M\). Since \(\Omega _t\) is a small perturbation of \(B_R\), we may assume that the assumption (1.3) holds, and so by (2.20) we have
Therefore, \(\Omega _t\), \(\mathbf{u}\) and \({\mathfrak {p}}\) satisfy equations (1.1), and so we see that Theorem 1 follows immediately from Theorem 4. \(\square \)
2.2 Two-phase problem
We now formulate problem (1.9) in the fixed domain. The idea is essentially the same as in the one-phase case. Let \(\dot{\Omega }= \Omega {\setminus } S_R\), \(\Omega _+=B_R\) and \(\Omega _-=\Omega {\setminus }\overline{B_R}\). We define the barycenter point, \(\xi (t)\), of \(\Omega _{+t}\) by setting
where we have used the fact that \(|\Omega _{+t}| = |B_R|\), which follows from the assumption (1.12). By the Reynolds transport theorem, we see that
Let \(\rho (y, t)\) be an unknown periodic function with period \(2\pi \) such that
where \(S_R = \{x \in {\mathbb R}^N \mid |x| = R\}\) and \(\mathbf{n}\) is the unit outer normal to \(S_R\), that is \(\mathbf{n}= y/|y|\) for \(y \in S_R\).
In the following, we fix the method how to extend this to a transformation from \(\dot{\Omega }\) to \(\Omega _t\). Let H be a unique solution of the Dirichlet problem:
Let L be a large number for which \(\Omega \subset B_L\). From the K-method in real interpolation theory [9, 21], we see that
for any \(t \in (0, 2\pi )\). We may assume that there exists a small number \(\omega > 0\) for which \(B_{R+3\omega } \subset \Omega \). Let \(\varphi \) be a function in \(C^\infty ({\mathbb R}^N)\) for which equals one for \(x \in B_{R+\omega }\) and zero for \(x\not \in B_{R+2\omega }\). Let \(\Phi (y, t) = y+ \varphi (y)(R^{-1}H_\rho (y, t)y+ \xi (t))\). Notice that \(\Phi (y, t) = y + R^{-1}H_\rho (y, t)y + \xi (t)\) for \(y \in B_R\). Setting \(\Psi (y, t) = \varphi (y)(R^{-1}H_\rho (y, t)y+ \xi (t))\), we assume that
with some small constant \(\delta > 0\). We choose \(\delta > 0\) so small that the map: \(y \mapsto x= \Phi (y, t)\) is bijective from \(\Omega \) onto itself. In fact, for any \(y_1\) and \(y_2\)
which leads to the injectivity of the map: \(x = \Phi (y, t)\) for any \(t \in {\mathbb R}\) provided that \(0< \delta < 1\). Moreover, using the fact that \(x = \Phi (y, t) = y\) for \(y \in \Omega {\setminus } B_{R+2\omega }\), and the inverse mapping theorem, we see that the map \(x=\Phi (y, t)\) is surjective from \(\Omega \) onto itself. Let
Notice that \(R^{-1}y\) is the unit outer normal to \(S_R\) for \(y \in S_R\). In the following, the jump quantity of f defined on \(\Omega {\setminus } S_R\) is also denoted by [[f]], which is defined by setting
where we have set \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R \cup S_R)\). Let \(\dot{\Omega }= \Omega _+ \cup \Omega _-\), and for f defined on \(\dot{\Omega }\), we write \(f_\pm = f|_{\Omega _\pm }\). On the other hand, for \(f_\pm \) defined on \(\Omega _\pm \), we define f by \(f|_{\Omega _\pm } = f_\pm \).
Let \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy the equations (1.9), and let \(\Phi ^{-1}(x, t)\) be the inverse map of \(x = \Phi (y, t)\). Let \(\mathbf{v}_\pm (y, t) = \mathbf{u}_\pm (\Phi ^{-1}(y, t), t)\) and \({\mathfrak {q}}_\pm (y, t) = {\mathfrak {p}}_\pm (\Phi ^{-1}(y, t), t)\) for \(y \in \Omega _{\pm t}\). We derive an equation for \(\mathbf{v}_+\) and \(\rho \) from the kinematic condition \(V_{\Gamma _t} = \mathbf{u}\cdot \mathbf{n}_t\) on \(\Gamma _t\). Noting that \([[\mathbf{u}]] = 0\) on \(\Gamma _t\), we may also assume that \([[\mathbf{v}]]=0\) on \(S_R\), and so \(\mathbf{v}_+ = \mathbf{v}_-\) on \(S_R\).
From the definition it follows that
Here and in the following, the unit outer normal to \(S_R\) is denoted by \(\mathbf{n}\), which is given by \(\mathbf{n}(y) = R^{-1}y\) for \(y \in S_R\). To represent the time derivative of \(\xi (t)\) given in (2.21), we introduce the Jacobian \(J_+(t)\) of the transformation: \(x = y + R^{-1}H_\rho y + \xi (t)\) for \(y \in B_R\), which is written as \(J_+(t) = 1 + J_{0,+}(t)\) with
Choosing \(\delta > 0\) small enough in (2.24), we have
From (2.21) it follows that
and noting that \(\mathbf{n}\cdot \mathbf{n}=1\), on \(S_R\) we have the kinematic equation:
with
As was already discussed in Sect. 2.1, from the assumption (1.12) and the representation formulas of \(\Omega _{+t}\) and \(\Gamma _t\) in (2.25), we have (2.12) in Sect. 2.1, too. Moreover, from (2.21) and the assumption (1.12), we have (2.13) in Sect. 2.1, too. Thus, under the assumption (1.12) and the representation of \(\Gamma _t\) and \(\Omega _{+t}\) in (2.25), the kinematic condition is equivalent to the equation:
with
And then, to prove Theorem 3, we shall prove the global well-posedness of the following equations:
where we have set
and \({\mathcal M}\rho \) and \({\mathcal B}_R\rho \) are the same as in (2.17) in Sect. 2.1. For the functions on the right side of equations (2.31), \(\mathbf{G}_\pm \) and \(\mathbf{F}_\pm (\mathbf{v}, \rho )\) are defined in (3.39) of Sect. 3, \(g_\pm (\mathbf{v}, \rho )\) and \(\mathbf{g}_\pm (\mathbf{v}, \rho )\) are defined in (3.38) of Sect. 3, and \(\tilde{\mathbf{h}}(\mathbf{v}, \rho )\) is defined in (3.40) of Sect. 3 .
The following theorem is the unique existence theorem of \(2\pi \)-periodic solutions of problem (2.31).
Theorem 5
Let \(1< p, q < \infty \) and \(2/p+N/q < 1\). Then, there exists a small constant \(\epsilon > 0\) such that for any \(\mathbf{f}\in L_{p, \mathrm{per}}((0, 2\pi ), L_q(\Omega )^N)\) satisfying the smallness condition: \(\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(\Omega ))} \le \epsilon \), problem (2.31) admits solutions \(\mathbf{v}_\pm \), \({\mathfrak {q}}_\pm \), and \(\rho \) satisfying the regularity condition (1.13) and the estimate (1.14) in Theorem 3.
Employing the same argument as in the proof of Theorem 1 in Sect. 2.1, we see that Theorem 3 immediately follows from Theorem 5.
3 Derivation of nonlinear terms
3.1 One-phase problem case
First, we consider the one-phase problem case and we consider the map
where \(\Psi (y, t)= R^{-1}H_\rho (y,t) y + \xi (t)\) and \(H_\rho \) satisfies the condition (2.5) and (2.6). Recall that \(H_\rho (y, t) = \rho (y, t)\) for \(y \in S_R\). Let \(\Omega _t\), \(\Gamma _t\), \(\mathbf{u}(x, t)\) and \({\mathfrak {p}}(x, t)\) satisfy the equations (1.1) and
Choose \(\delta > 0\) small in such a way that there exists an inverse map: \(y = \Phi ^{-1}(x, t)\) of the map: \(x = \Phi (y,t)=y+\Psi (y, t)\). Let \(\mathbf{v}(y, t) = \mathbf{u}(\Phi ^{-1}(y, t), t)\) and \({\mathfrak {q}}(y, t) = {\mathfrak {p}}(\Phi ^{-1}(y, t), t)\). By the chain rule, we have
where \(\nabla _z = {}^\top (\partial /\partial z_1, \ldots , \partial /\partial z_N)\) for \(z \in \{x, y\}\) and \(\mathbf{k}=(k_0, k_1, \ldots , k_N) = (H_\rho , \nabla H_\rho )\). Here, \(\mathbf{V}_0(\mathbf{k})\) is an \((N\times N)\)-matrix of \(C^\infty \) functions defined for \(|\mathbf{k}| \le \delta \) with \(\mathbf{V}_0(0) = 0\) and \(V_{0ij}(\mathbf{k})\) is the \((i, j)\mathrm{th}\) component of \(\mathbf{V}_0(\mathbf{k})\). By (3.2), we can write \(\mathbf{D}(\mathbf{u})\) as \(\mathbf{D}(\mathbf{u}) = \mathbf{D}(\mathbf{v}) + {\mathcal D}_\mathbf{D}(\mathbf{k})\nabla \mathbf{v}\) with
We next consider \(\mathrm{div}\,\mathbf{v}\). By (3.2), we have
Let J be the Jacobian of the transformation (3.1). Choosing \(\delta > 0\) small enough, we may assume that \(J = J(\mathbf{k}) = 1 + J_0(\mathbf{k})\), where \(J_0(\mathbf{k})\) is a \(C^\infty \) function defined for \(|\mathbf{k}| < \sigma \) such that \(J_0(0) = 0\).
To obtain another representation formula of \(\mathrm{div}\,_x\mathbf{u}\), we use the inner product \((\cdot , \cdot )_{\Omega _t}\). For any test function \(\varphi \in C^\infty _0(\Omega _t)\), we set \(\psi (y) = \varphi (x)\). We then have
which, combined with (3.4), leads to
Recalling that \(J = J(\mathbf{k}) = 1 + J_0(\mathbf{k})\), we define \(g(\mathbf{v}, \rho )\) and \(\mathbf{g}(\mathbf{v}, \rho )\) by letting
and then by (3.5) we see that the divergence free condition: \(\mathrm{div}\,\mathbf{u}=0\) is transformed to the second equation in the equations (2.16). In particular, it follows from (3.5) that
To derive \(\mathbf{F}(\mathbf{v}, \rho )\), we first observe that
where we have used (3.3). Since
we have
and therefore,
Putting (3.8) and (3.9) together gives
Since \((\mathbf{I}+ \nabla \Psi )(\mathbf{I}+ \mathbf{V}_0) = (\partial x/\partial y)(\partial y/\partial x) = \mathbf{I}\), that is,
we have
Thus, changing i to \(\ell \) and m to i in the formula above, we define an N-vector of functions \(\mathbf{F}_1(\mathbf{v}, \rho )\) by letting
where \(\mathbf{F}_1(\mathbf{u}, \rho )|_i\) denotes the \(i\mathrm{th}\) component of \(\mathbf{F}_1(\mathbf{u}, \rho )\).
Moreover,
with
where we have set
Thus, setting
we have the first equation in equations (2.16).
We next consider the transformation of the boundary conditions. Recall that \(\Gamma _t\) is represented by \(x = y + \rho (y, t)\mathbf{n}(y) + \xi (t)\) for \(y \in S_R\) with \(\mathbf{n}(y)=y/|y|\). Let \(x_0\) be any point on \(S_R\) and let \(\Phi (p)\) be a \(C^\infty \) diffeomorphism on \({\mathbb R}^N\) such that—up to a rotation—it holds
where we have set \(B_\omega (x_0) = \{y \in {\mathbb R}^N \mid |y - x_0| < \omega \}\) and \(p' = (p_1, \ldots , p_{N-1})\). Notice that \(y = \Phi (p', 0) \in S_R \cap B_\omega (x_0)\) and \(\rho (y, t) = H_\rho (\Phi (p',0), t)\). Let \(\{x_k\}_{k=1}^K\) and \(\{\zeta _k\}_{k= 1}^K\) be a finite number of points on \(S_R\) and a partition of unity of \(S_R\) such that \(\mathrm{supp}\, \zeta _k \subset B_\omega (x_k)\) and \(\sum _{k=1}^K\zeta _k(y) = 1\) on \(S_R\). In the following, we represent functions on each \(S_R\cap B_\omega (x_k)\), and to represent functions globally, we use the formula:
Thus, for the detailed calculations, we only consider the domain \(B_R \cap B_\omega (x_\ell )\) (\(\ell =1, \ldots , K\)), and use the local coordinate system: \(y = \Phi _\ell (p)\) for \(p \in U\), where we have written \(\Phi =\Phi _\ell \), and \(U = \{p \in {\mathbb R}^N \mid 0< p_N< \omega , |p'| < \omega \}\).
We write \(\rho = \rho (y(p_1, \ldots , p_{N-1}, 0), t)\) in the following. By the chain rule, we have
where we have set \(\Phi _\ell = {}^\top (\Phi _{\ell ,1}, \ldots , \Phi _{\ell ,N})\), and so, \(\partial \rho /\partial p_i\) is defined in \(B_\omega (x_0)\) by letting
We first represent \(\mathbf{n}_t\). Since \(\Gamma _t\) is given by \(x = y + \rho (y, t)\mathbf{n}+ \xi (t)\) for \(y \in S_R\),
The vectors \(\tau _i\) (\(i=1, \ldots , N-1\)) form a basis of the tangent space of \(S_R\) at \(y=y(p_1,\ldots , p_{N-1})\). Since \(|\mathbf{n}_t|^2 =1\), we have
because \(\tau _i\cdot \mathbf{n}=0\). The vectors \(\dfrac{\partial x}{\partial p_i}\) \((i=1, \ldots , N-1)\) form a basis of the tangent space of \(\Gamma _t\), and so \(\mathbf{n}_t\cdot \dfrac{\partial x}{\partial p_i}=0\). Thus, we have
Since \(\mathbf{n}\cdot \dfrac{\partial y}{\partial p_i} = \mathbf{n}\cdot \tau _i= 0\), \(\dfrac{\partial \mathbf{n}}{\partial p_i}\cdot \mathbf{n}= 0\) (because of \(|\mathbf{n}|^2=1\)), and \(\dfrac{\partial y}{\partial p_i}\cdot \dfrac{\partial y}{\partial p_j} = \tau _i\cdot \tau _j=g_{ij}\), recalling that \(\mathbf{n}= R^{-1}y= R^{-1}\Phi _\ell \), by (3.18) we have
Let \(G=(g_{ij})\) and \(G^{-1} = (g^{ij})\), and then setting \(\nabla _\Gamma '\rho = (\partial \rho /\partial p_1, \ldots , \partial \rho /\partial p_{N-1})\), we have
which leads to
Moreover, combining (3.17) and (3.19), we have
Using the formula:
we have
with
Combining these formulas obtained above gives
where we have set
From (3.16), \(\nabla '_\Gamma \rho \) is extended to \({\mathbb R}^N\) by the formula: \(\nabla '_\Gamma \rho = (\nabla \Phi _\ell )\nabla \Psi _\rho \circ \Phi _\ell \), and so we may write
on \(B_\omega (x_\ell )\) with some function \(\mathbf{V}_{\mathbf{n}, \ell }(\mathbf{k}) = \mathbf{V}_{\mathbf{n},\ell }(y, \mathbf{k})\) defined on \(B_\omega (x_\ell )\times \{\mathbf{k}\mid |\mathbf{k}| \le \delta \}\) with \(\mathbf{V}_{\mathbf{n}, \ell }(0) = 0\) possessing the estimate
with some constant C independent of \(\ell \). Here and in the following \(\mathbf{k}\) are the variables corresponding to \(\bar{\nabla }H_\rho = (H_\rho , \nabla H_\rho )\). In view of (3.21), we have
Thus, in view of (3.14) and (3.16), we may write
where \(\partial '_j\rho = \partial \rho /\partial p_j\) locally on \(B_\omega (x_\ell ) \cap S_R\), \(\bar{\nabla }H_\rho = (H_\rho , \nabla H_\rho )\), and \(\mathbf{V}_{\mathbf{n}}(\mathbf{k})\) is a matrix of functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:
And also we may write
where \(\tilde{\mathbf{V}}_{\mathbf{n}}(\mathbf{k})\) is a matrix of functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:
We now consider the boundary condition:
It is convenient to divide the formula in (3.27) into the tangential part and normal part on \(\Gamma _t\) as follows:
Here, \(\Pi _t\) is defined by \(\Pi _t\mathbf{d}= \mathbf{d}- < \mathbf{d}, \mathbf{n}_t>\mathbf{n}_t\) for any N-vector of functions \(\mathbf{d}\). In the last equation in equations (2.16), we set \(\mathbf{h}'(\mathbf{v}, \rho ) = \mathbf{h}(\mathbf{v}, \rho ) - <\mathbf{h}(\mathbf{v}, \rho ), \mathbf{n}>\mathbf{n}\) and \(h_N(\mathbf{v}, \rho ) = <\mathbf{h}(\mathbf{v}, \rho ), \mathbf{n}>\). By (3.25) and (3.3), we see that the boundary condition (3.28) is transformed to the following formula:
where we have set \(\mathbf{d}_\tau = \mathbf{d}- <\mathbf{d}, \mathbf{n}>\mathbf{n}\) and
Finally, we derive the nonlinear term \(h_N(\mathbf{u}, \rho )\) in (3.29). Recall that \(\Gamma _t\) is represented by \(x = (R + \rho )\mathbf{n}(y) + \xi (t)\) for \(y \in S_R\), where \(\mathbf{n}= y/|y| \in S_1\). Then, we have
where \(\tau _j = \frac{\partial \mathbf{n}}{\partial p_j}\), which forms a basis of the tangent space of \(S_1\). Since \(\tau _j\cdot \mathbf{n}= 0\), the \((i, j)\mathrm{th}\) component of the first fundamental form \(G_t=(g_{tij})\) of \(\Gamma _t\) is given by
where \(g_{ij} = \tau _i\cdot \tau _j\) is the \((i, j)\mathrm{th}\) element of the first fundamental form, G, of \(S_1\), and so
Since
for any \((N-1)\)-vectors \(\mathbf{a}'\) and \(\mathbf{b}' \in {\mathbb R}^{N-1}\), we have
Here and in the following, \(O_2\) denotes a symbol defined by setting
with some coefficients \(a_0\), \(b_j\) and \(c_{ij}\) defined on \(\overline{B_R}\) satisfying the estimate: \(|(a_0, b_j, c_{ij})(y, t)| \le C\) and \(|\nabla (a_0, b_j, c_{ij})(y, t)| \le C|\nabla ^2 H_\rho (y, t)|\) provided that \(\Vert H_\rho \Vert _{L_\infty ((0, 2\pi ), H^1_\infty (B_R))} \le \delta \). In particular,
componentwise.
We next calculate the Christoffel symbols of \(\Gamma _t\). Since
we have
where \(\ell _{ij} = <\tau _{ij}, \mathbf{n}>\), and so
Thus,
where \(\bar{\nabla }'^2f\) is an \(((N-1)^2+N)\)-vector of the form: \(\bar{\nabla }'^2f = (\partial _i\partial _jf, \partial _if, f \mid i, j=1, \ldots , N-1)\), \(\partial _i = \partial /\partial p_i\), \(\nabla '^2_p = (\partial _i\partial _j\rho \mid i, j=1, \ldots , N-1)\), and
and so
Combining this formula with (3.21), using \(<\partial _i\mathbf{n}, \mathbf{n}> = 0\), \(<\mathbf{n}, \tau _\ell > = 0\), \(\Delta _{S_1}\mathbf{n}= -(N-1)\mathbf{n}\), and (3.15) gives
where \(O_1\) denotes a symbol defined by setting
with some coefficients \(a_0'\) and \(b_j'\) defined on \(\overline{B_R}\) satisfying the estimate: \(|(a_0', b_j')(y, t)|\le C\) and \(|\nabla (a_0', b_j')(y, t)| \le C|\nabla ^2H_\rho (y, t)|\) provided that \(\Vert H_\rho \Vert _{L_\infty ((0, 2\pi ), H^1_\infty (B_R))} \le \delta \). Since
we have
Setting \(p_0 = -(N-1)/R\), from (3.27) we have
on \(S_R\times (0, 2\pi ) \). Here, in view of (3.3) and (3.33), we have defined \(h_N(\mathbf{v}, \rho )\) by letting
where \(\mathbf{V}_{h, N}(\mathbf{k})\) and \(\tilde{\mathbf{V}}'_\Gamma (\mathbf{k})\) are functions defined on \(\overline{B_R}\times \{\mathbf{k}\mid |\mathbf{k}| < \delta \}\) possessing the estimate:
for some constant C.
3.2 Two-phase problem case
Let \(\Omega _+ = B_R\) and \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\). In the two-phase case, we let
Let \(J_\pm (t)\) be the Jacobian of the map: \(x = y+ \Psi _\pm (y, t)\) for \(y \in \Omega _\pm \), which are defined by setting
Notice that
where c is the unique constant for which the following equality holds:
We assume that
with suitably small constant \(\delta > 0\). Since
there exists a constant \(\delta _1 > 0\) such that if
then the condition for \(\xi (t)\) in (3.36) holds. Thus, in the proof of Theorem 5, we assume that the conditions (3.36) and (3.37) hold.
Set \(J_{0\pm }(t) = J_\pm (t) - 1\). By the chain rule, we have
where \(\mathbf{V}_{\pm 0}(\mathbf{k}_\pm )\) is given by
Here and in the following, \(\mathbf{k}_+\) and \(\mathbf{k}_-\) denote the variables corresponding to \((H_\rho , \nabla H_\rho )\) and \((\Psi _{-, \rho }, \nabla \Psi _{-, \rho })\).
Employing the same argument as for obtaining the formulas in (3.6), we have
And also, from (3.13) we have
with
Here and in the following, we have set \(\Psi _{\pm }(y, t) = {}^\top (\Psi _{\pm 1}(y, t), \ldots , \Psi _{\pm N}(y, t))\), \(\mathbf{v}_\pm = {}^\top (v_{\pm 1}, \ldots , v_{\pm N})\), and
To define the right hand side of the transmission condition, we use (3.31) and (3.34). We first introduce a symbol \(((\cdot ))\). For \(f_\pm \), let \([f_\pm ]\) be a suitable extension of \(f_\pm \) to \(\Omega _{\mp }\) such that
with some constant \(C_k\). Here, if the right-hand side is finite, then \([f_\pm ]\) and \(\partial _t[f_\pm ]\) exist and the estimates above hold. In particular, we set \(H^0_q(\Omega _\pm ) = L_q(\Omega _\pm )\). We set
And then, ((f)) is defined by setting
Using this symbol, we can proceed as for the derivation of (3.31) and (3.34) and define \(\tilde{\mathbf{h}}'(\mathbf{v}, \rho )\) and \(\tilde{h}_N(\mathbf{v}, \rho )\) by setting
And then, we set \(\tilde{\mathbf{h}}(\mathbf{v}, \rho ) =(\tilde{\mathbf{h}}'(\mathbf{v}, \rho ), \tilde{h}_N(\mathbf{v}, \rho ))\).
4 On periodic solutions of the linearized equations
In this section, we shall prove the \(L_p\)–\(L_q\) maximal regularity of \(2\pi \)-periodic solutions of the linearized equations.
4.1 On linearized problem of one-phase problem
In this subsection, we consider the \(L_p\)-\(L_q\) maximal regularity of periodic solutions to linearized equations:
where \({\mathcal L}\), \({\mathcal M}\), and \({\mathcal A}\) are the linear operators defined in (2.17). We shall prove the unique existence theorem of \(2\pi \)-periodic solutions of equations (4.1). Our main result is this section is stated as follows.
Theorem 6
Let \(1< p, q < \infty \). Then, for any \(\mathbf{F}\), D, G, \(\mathbf{G}\) and \(\mathbf{H}\) with
problem (4.1) admits unique solutions \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) with
possessing the estimate:
for some constant \(C > 0\).
To prove Theorem 6, our approach is to use the \({\mathcal R}\)-solver, Weis’ operator-valued Fourier multiplier theorem [22] and a transference theorem, which is created in Eiter, Kyed and Shibata [2]. To introduce the notion of \({\mathcal R}\)-solver, we introduce the \({\mathcal R}\)-boundedness of operator families.
Definition 7
Let X and Y be two Banach spaces. A family of operators \({\mathcal T}\subset {\mathcal L}(X, Y)\) is called \({\mathcal R}\)-bounded on \({\mathcal L}(X, Y)\), if there exist a constant \(C > 0\) and \(p \in [1, \infty )\) such that for each \(n \in {\mathbb N}\), \(\{T_j\}_{j=1}^n \in {\mathcal T}^n\), and \(\{f_j\}_{j=1}^n \in X^n\), we have
Here, the Rademacher functions \(r_k\), \(k \in {\mathbb N}\), are given by \(r_k : [0, 1] \rightarrow \{-1, 1\}\), \(t \mapsto \mathrm{sign}\,(\sin 2^k\pi t)\). The smallest such C is called \({\mathcal R}\)-bound of \({\mathcal T}\) on \({\mathcal L}(X, Y)\), which is denoted by \({\mathcal R}_{{\mathcal L}(X, Y)}{\mathcal T}\).
We quote Weis’ operator-valued Fourier multiplier theorem and the transference theorem for operator-valued Fourier multipliers.
Theorem 8
[Weis] Let X and Y be two UMD Banach spaces. Let \(m \in C^1({\mathbb R}{\setminus }\{0\}, {\mathcal L}(X, Y))\) satisfies the multiplier condition:
for \(\ell = 0, 1\) with some constant \(r_b\). Let \(T_m\) be a multiplier defined by \(T_m[f] = {\mathcal F}^{-1}[m{\mathcal F}[f]]\). Then, \(T_m \in {\mathcal L}(L_p({\mathbb R}, X), L_p({\mathbb R}, Y))\) with
for any \(p \in (1, \infty )\) with some constant \(C_p\) depending on p but independent of \(r_b\).
The transference theorem for operator-valued Fourier multipliers obtained in [2] is stated as follows.
Theorem 9
Let X and Y be two Banach spaces and \(p \in (1, \infty )\). Assume that Y is reflexive. Let
and let \(m|_{\mathbb T}\) denote the restriction of m on \({\mathbb T}\). We define multipliers on \({\mathbb R}\) and \({\mathbb T}\) associated with m by setting
If \(T_{m, {\mathbb R}} \in {\mathcal L}(L_p({\mathbb R}, X), L_p({\mathbb R}, Y))\) possessing the estimate:
for any \(f \in L_p({\mathbb R}, X)\) with some constant M, then \(T_{m, {\mathbb T}} \in {\mathcal L}(L_p({\mathbb T}, X), L_p({\mathbb T}, Y))\) and
for any \(f \in L_{p}({\mathbb T}, X)\) with some constant \(C_p\) depending solely on p and independent of M.
Remark 10
In the usual scalar-valued multiplier case, the transference theorem was proved by de Leeuw [1], and so this theorem is an extension to the operator-valued case.
We now consider the \({\mathcal R}\)-solver of the generalized resolvent problem:
for \(k \in {\mathbb R}\). From Theorem 4.8 in Shibata [18] (cf. also Shibata [15, 16]) we know the following theorem concerned with the existence of an \({\mathcal R}\)-solver of problem (4.1).
Theorem 11
Let \(1< q < \infty \) and let \({\mathbb R}_{k_0} = {\mathbb R}{\setminus }(-k_0, k_0)\). Let
Then, there exist a constant \(k_0 > 0\) and operator families \({\mathcal A}(ik)\), \({\mathcal P}(ik)\), and \({\mathcal H}(ik)\) with
such that for any \((\mathbf{f}, d, \mathbf{h}, g, \mathbf{g})\) and \(k \in {\mathbb R}_{k_0}\), \(\mathbf{v}= {\mathcal A}(ik){\mathcal F}_k\), \({\mathfrak {q}}= {\mathcal P}(ik){\mathcal F}_k\) and \(\eta = {\mathcal H}(ik){\mathcal F}_k\), where
are unique solutions of equations (4.3), and
for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\) with some constant \(r_b\).
Remark 12
-
(1)
Here and in the following, for \(\theta \in (0,1)\) we set
$$\begin{aligned}(ik)^\theta = {\left\{ \begin{array}{ll} \mathrm{e}^{i\pi \theta /2}| k|^\theta &{}\quad \text {for }k > 0, \\ \mathrm{e}^{-i\pi \theta /2}| k|^\theta &{}\quad \text {for }k < 0. \end{array}\right. } \end{aligned}$$ -
(2)
The functions \(F_1\), \(F_2\), \(F_3\), \(F_4\), \(F_5\), \(F_6\), and \(F_7\) are variables corresponding to \(\mathbf{f}\), d, \((ik)^{1/2}\mathbf{h}\), \(\mathbf{h}\), \((ik)^{1/2}g\), g, and \(ik\, \mathbf{g}\), respectively.
-
(3)
We define the norm \(\Vert \cdot \Vert _{{\mathcal X}_q(B_R)}\) by setting
$$\begin{aligned}&\Vert (F_1, \ldots , F_7)\Vert _{{\mathcal X}_q(B_R)} = \Vert (F_1, F_3, F_5, F_7)\Vert _{L_q(B_R)}\\&\qquad + \Vert F_2\Vert _{W^{2-1/q}_q(S_R)} + \Vert (F_4, F_6)\Vert _{H^1_q(B_R)}. \end{aligned}$$
Let \(\varphi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+2}\) and zero for \(k \not \in {\mathbb R}_{k_0+1}\), and let \(\psi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+4}\) and zero for \(k \not \in {\mathbb R}_{k_0+3}\). Notice that \(\varphi (ik)\psi (ik) = \varphi (ik)\). Let \({\mathcal A}(ik)\), \({\mathcal P}(ik)\) and \({\mathcal H}(ik)\) be the \({\mathcal R}\)-solvers given in Theorem 11. Then, we have
for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\). To prove (4.5), we use the following lemma concerning the fundamental properties of the \({\mathcal R}\)-bound and scalar-valued Fourier multipliers.
Lemma 13
(a) Let X and Y be Banach spaces, and let \({\mathcal T}\) and \({\mathcal S}\) be \({\mathcal R}\)-bounded families in \({\mathcal L}(X,Y)\). Then, \({\mathcal T}+ {\mathcal S}= \{T + S \mid T \in {\mathcal T}, S \in {\mathcal S}\}\) is also an \({\mathcal R}\)-bounded family in \({\mathcal L}(X,Y)\) and
(b) Let X, Y and Z be Banach spaces, and let \({\mathcal T}\) and \({\mathcal S}\) be \({\mathcal R}\)-bounded families in \({\mathcal L}(X, Y)\) and \({\mathcal L}(Y, Z)\), respectively. Then, \({\mathcal S}{\mathcal T}= \{ST \mid T \in {\mathcal T}, S \in {\mathcal S}\}\) is also an \({\mathcal R}\)-bounded family in \({\mathcal L}(X, Z)\) and
(c) Let \(1< p, \, q < \infty \) and let D be a domain in \({\mathbb R}^N\). Let \(m=m(\lambda )\) be a bounded function defined on a subset U of \({\mathbb C}\) and let \(M_m(\lambda )\) be a map defined by \(M_m(\lambda )f = m(\lambda )f\) for any \(f \in L_q(D)\). Then, \({\mathcal R}_{{\mathcal L}(L_q(D))}(\{M_m(\lambda ) \mid \lambda \in U\}) \le C_{N,q,D}\Vert m\Vert _{L_\infty (U)}\).
(d) Let \(n=n(\tau )\) be a \(C^1\)-function defined on \({\mathbb R}{\setminus }\{0\}\) that satisfies the conditions \(|n(\tau )| \le \gamma \) and \(|\tau n'(\tau )| \le \gamma \) with some constant \(c > 0\) for any \(\tau \in {\mathbb R}{\setminus }\{0\}\). Let \(T_n\) be an operator-valued Fourier multiplier defined by \(T_n f = {\mathcal F}^{-1}[n {\mathcal F}[f]]\) for any f with \({\mathcal F}[f] \in {\mathcal D}({\mathbb R}, L_q(D))\). Then, \(T_n\) is extended to a bounded linear operator from \(L_p({\mathbb R}, L_q(D))\) into itself. Moreover, denoting this extension also by \(T_n\), we have
Here, we only prove the \({\mathcal R}\)-boundedness of \(\varphi (ik)ik{\mathcal A}(ik)\). The \({\mathcal R}\)-boundedness of the other terms can be proved by the same argument. Let \(n \in {\mathbb N}\), \(\{k_\ell \}_{\ell =1}^n \in {\mathbb R}^n\), \(\{F_\ell \}_{\ell =1}^n \in {\mathcal X}_q(B_R)^n\). Changing the labeling of indices if necessary, we may assume that \(\varphi (k_\ell )\not =0\) for \(k = 1, \ldots , m\) and \(\varphi (k_\ell ) = 0\) for \(\ell =m+1,\ldots , n\). And then, using Lemma 13, we have
which shows that
For \(f \in \{\mathbf{F}, G, \mathbf{G}, D, \mathbf{H}\}\), let
We consider the high frequency part of the equations (4.1):
By Theorem 8, Theorem 9, and (4.5), we have immediately the following theorem.
Theorem 14
Let \(1< p, q < \infty \). Then, for any functions \(\mathbf{F}\), G, \(\mathbf{G}\), D, and \(\mathbf{H}\) with
We let
where we have set
Then, \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) are the unique solutions of equations (4.6), which possess the following estimate:
for some constant \(C > 0\). Here, we have set
We now consider the lower frequency part of solutions of equations (4.1). Namely, we consider equations (4.3) for \(k \in {\mathbb R}\) with \(1 \le |k| < k_0+4\). We shall show the following theorem.
Theorem 15
Let \(1< q < \infty \) and \(k \in {\mathbb Z}\) with \(1 \le |k| \le k_0+3\). Then, for any \(\mathbf{f}\in L_q(B_R)^N\), \(g \in H^1_q(B_R)\), \(d \in W^{2-1/q}_q(S_R)\), \(\mathbf{h}\in H^1_q(B_R)^N\), and \(\mathbf{g}\in L_q(B_R)^N\), problem (4.3) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {q}}\in H^1_q(B_R)\), and \(\eta \in W^{3-1/q}_q(S_R)\) possessing the estimate:
for some constant \(C > 0\).
Proof
From Theorem 11, problem (4.3) with \(k = k_0+4\) admits unique solutions \(\mathbf{v}_{k_0} \in H^2_q(B_R)^N\), \({\mathfrak {q}}_{k_0} \in H^1_q(B_R)\), and \(\eta _{k_0} \in W^{3-1/q}_q(S_R)\) possessing the estimate:
for some constant C. Thus, for any \(k \in {\mathbb R}\) with \(|k| < k_0+4\), we consider the unique solvability of the equations:
where we have set \(\mathbf{f}= i(k-k_0)\mathbf{v}_{k_0}\) and \(d=i(k_0-k)\eta _{k_0}\). In fact, if we set \(\mathbf{v}= \mathbf{v}_{k_0}+\mathbf{w}\), \({\mathfrak {q}}={\mathfrak {q}}_{k_0}+{\mathfrak {r}}\), and \(\eta = \eta _{k_0}+ \zeta \), then \(\mathbf{v}\), \({\mathfrak {q}}\) and \(\eta \) are unique solutions of equations (4.3).
In what follows, we study the unique solvability of equations (4.9) in the case where \(\mathbf{f}\in L_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\) are arbitrary. To solve (4.9), it is convenient to study the functional analytic form of (4.9), and so we eliminate the pressure term \({\mathfrak {r}}\) and the divergence condition \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Given \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\), let \(K=K(\mathbf{v}, \zeta ) \in H^1_q(B_R)\) be the unique solution of the weak Dirichlet problem:
subject to
where we have set
and \(q' = q/(q-1)\). In view of Poincaré’s inequality, \(\hat{H}^1_{q',0}(B_R) = H^1_{q', 0}(B_R) = \{\varphi \in H^1_{q'}(B_R) \mid \varphi |_{S_R}=0\}\). Instead of (4.9), we consider the equations:
In view of the boundary condition (4.11) for \(K(\mathbf{w}, \zeta )\), that \(\mathbf{w}\) and \(\zeta \) satisfy the third equation of equations (4.12) is equivalent to
where \(\mathbf{d}_\tau = \mathbf{d}- <\mathbf{d}, \mathbf{n}>\mathbf{n}\) for any N-vector \(\mathbf{d}\). Let \(J_q(B_R)\) be a solenoidal space defined by setting
Obviously, for \(\mathbf{v}\in H^1_q(B_R)\), in order that \(\mathrm{div}\,\mathbf{v}= 0\) in \(B_R\), it is necessary and sufficient that \(\mathbf{v}\in J_q(B_R)\). For any \(\mathbf{f}\in L_q(B_R)^N\), let \(\psi \in H^1_{q,0}(B_R)\) be a unique solution of the weak Dirichlet problem:
Let \(\mathbf{g}=\mathbf{f}-\nabla \psi \) and inserting this formula into equations (4.9), we have
where we have used the fact that \(\psi |_{S_R}=0\). Therefore, we shall solve equations (4.9) for \(\mathbf{f}\in J_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\). When \(\mathbf{f}\in J_q(B_R)\), the equations (4.9) and (4.12) are equivalent. In fact, if \(\mathbf{w}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\) satisfy equations (4.9) with some \({\mathfrak {r}}\in H^1_q(B_R)\). Then, for any \(\varphi \in \hat{H}^1_{q',0}(B_R)\), we have
where we have used the fact that \(\mathrm{div}\,\mathbf{w}= 0\). Moreover, from the boundary conditions in equations (4.9) and (4.11), it follows that
on \(S_R\) because \(\mathrm{div}\,\mathbf{w}=0\). Thus, the uniqueness of the solutions to his weak Dirichlet problem yields that \({\mathfrak {r}}= K(\mathbf{w}, \zeta )\), and so \(\mathbf{w}\) and \(\zeta \) satisfy equations (4.12). Conversely, let \(\mathbf{w}\in H^2_q(B_R)^N\) and \(\zeta \in W^{3-1/q}_q(S_R)\) be solutions of equations (4.12). For any \(\varphi \in \hat{H}^1_{q', 0}(B_R)\), we have
Moreover, from the boundary condition (4.13) it follows that \(\mathrm{div}\,\mathbf{w}=0\) on \(S_R\). The uniqueness implies that \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Thus, \(\mathbf{w}\), \({\mathfrak {r}}= K(\mathbf{w}, \zeta )\) and \(\zeta \) are solutions of equations (4.9). In particular, for solutions \(\mathbf{w}\) and \(\zeta \) of equations (4.12), we see that \(\mathbf{w}\) satisfies the divergence condition: \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\) automatically.
From now on, we study the unique existence theorem for equations (4.12) for any \(\mathbf{f}\in J_q(B_R)\) and \(d\in W^{2-1/q}_q(S_R)\). To formulate problem (4.12) in a functional analytic setting, we define the spaces \({\mathcal H}_q\), \({\mathcal D}_q\) and the operator \(\mathbf{A}\) by setting
where we have used (4.13) and \(\mathrm{div}\,\mathbf{w}=0\) in the definition of \({\mathcal D}_q\). We write equations (4.12) as
In view of Theorem 11, we see that \(k=k_0+4\) is an element of the resolvent set of the operator \(\mathbf{A}\), and so \((i(k_0+4)\mathbf{I}- \mathbf{A})^{-1}\) exists in \({\mathcal L}({\mathcal H}_q, {\mathcal D}_q)\). Since \(B_R\) is a compact set, it follows from the Rellich compactness theorem that \((i(k_0+4)\mathbf{I}- \mathbf{A})^{-1}\) is a compact operator from \({\mathcal H}_q\) into itself. Thus, in view of Riesz–Schauder theory, in particular, Fredholm alternative principle, that k belongs to the resolvent set if and only if uniqueness holds for k. Thus, our task is to prove the uniqueness of solutions to equations (4.14). Let \(U = (\mathbf{w}, \zeta ) \in {\mathcal D}_q\) satisfy the homogeneous equations:
Namely, \((\mathbf{w}, \zeta ) \in {\mathcal D}_q\) satisfies equations:
We first prove that
Integrating the second equation of equations (4.16) and applying the divergence theorem of Gauss gives that
where we have set \(|S_R| = \int _{S_R}\,\mathrm{d}\omega \) and we have used the fact that \(\mathrm{div}\,\mathbf{w}=0\) in \(B_R\). Thus, we have \((\zeta , 1)_{S_R} = 0\). Multiplying the second equation of equations (4.16) with \(x_j\), integrating the resultant formula over \(S_R\) and using the divergence theorem of Gauss gives that
because \((x_j, x_\ell )_{S_R}=0\) for \(j\not =\ell \). Since
we have \((\zeta , x_\ell )_{S_R} = 0\), because \((x_\ell , x_\ell )_{S_R} = (R^2/N)|S_R| > 0\). Thus, we have proved (4.17). In particular, \({\mathcal M}\zeta = 0\) in (4.16).
We now prove that \(\mathbf{w}= 0\). For this purpose, we first consider the case where \(2 \le q < \infty \). Since \(B_R\) is bounded, \({\mathcal D}_q \subset {\mathcal D}_2\). Multiplying the first equation of (4.16) with \(\mathbf{w}\) and integrating the resultant formula over \(B_R\) and using the divergence theorem of Gauss gives that
because \(\mathrm{div}\,\mathbf{w}= 0\) in \(B_R\). By the second equation of (4.16) with \({\mathcal M}\zeta =0\), we have
where we have used \(\mathbf{n}=R^{-1}x= R^{-1}(x_1,\ldots , x_N)\) for \(x \in S_R\). Thus,
Moreover, since \(\zeta \) satisfies (4.17), we know that
for some positive constant c, and therefore (4.18) implies \(\mathbf{w}=0\).
Now the first equation of (4.16) yields \(\nabla K(\mathbf{w},\zeta )=0\), so that \(K(\mathbf{w},\zeta )\) is constant. Integration of the third equation of (4.16) over \(S_R\) combined with (4.17) shows that this constant is 0, that is, \(K(\mathbf{w},\zeta )=0\).
Finally, the third equation of (4.16) yields that \({\mathcal B}_R\zeta =0\) on \(S_R\), and so by (4.17) we have \(\zeta =0\). This completes the proof of the uniqueness in the case where \(2 \le q < \infty \). In particular, we have the unique existence theorem of solutions to equation (4.14).
We now consider the case where \(1< q < 2\). Let \(\mathbf{f}\) be any element in \(J_{q'}(B_R)\) and let \(V = (\mathbf{v}, \eta ) \in {\mathcal D}_{q'}\) be a solution of the equation:
The existence of such V has already been proved above. Since \(d=0\), we see that \(\eta \) satisfies the relations:
and so \({\mathcal M}\eta = 0\). Using the divergence theorem of Gauss, we have
Using the fact that \(({\mathcal B}_R\zeta , x_j)_{S_R} = (x_j, {\mathcal B}_R\eta )_{S_R} =0\), we have
For any \(\mathbf{g}\in L_{q'}(B_R)^N\), let \(\psi \in \hat{H}^1_{q', 0}(B_R)\) be a unique solution of the weak Dirichlet problem:
Let \(\mathbf{f}= \mathbf{g}- \nabla \psi \), and then \(\mathbf{f}\in J_{q'}(B_R)\), and so using the fact that \(\mathbf{w}\in J_q(B_R)\), we have \((\mathbf{w}, \mathbf{g})_{B_R} = (\mathbf{w}, \mathbf{f})_{B_R} + (\mathbf{w}, \nabla \psi )_{B_R} = 0\). The arbitrariness of \(\mathbf{g}\in L_{q'}(B_R)^N\) implies that \(\mathbf{w}=0\). Thus, the second equation of (4.16) and (4.17) leads to \(\zeta =0\). This completes the proof of the uniqueness in the case where \(1< q < 2\), and therefore the proof of Theorem 15. \(\square \)
We now consider the linearized stationary problem:
We shall prove the following theorem.
Theorem 16
Let \(1< q < \infty \). Then, for any \(\mathbf{f}\in L_q(B_R)^N\), \(d \in W^{2-1/q}_q(S_R)\), \(g \in H^1_q(B_R)\), \(\mathbf{g}\in L_q(B_R)^N\), and \(\mathbf{h}\in H^1_q(B_R)^N\), problem (4.19) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\), and \(\rho \in W^{3-1/q}_q(S_R)\) possessing the estimate:
for some constant \(C > 0\).
Proof
The strategy of the proof is the same as that of Theorem 15. Since \({\mathcal L}\mathbf{v}\), \({\mathcal M}\rho \), and \(|B_R|^{-1}\int _{B_R}\mathbf{v}\,\mathrm{d}y\) are lower order perturbations, choosing \(k_0 > 0\) large enough, the generalized resolvent problem:
admits unique solutions: \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\), and \(\rho \in W^{3-1/q}_q(S_R)\) possessing the estimate (4.20). Of course, the constant C in (4.20) depends on \(k_0\) in this case, but \(k_0\) is fixed, and so we can say that C in (4.20) is some fixed constant. The essential part of the proof is to show the unique existence of solutions to equations (4.19) with \(g=\mathbf{g}=\mathbf{h}=0\), that is
And then, the uniqueness of the reduced problem in the \(L_2\) framework implies the unique existence of solutions as was studied in the proof Theorem 15. Thus, we define the reduced problem corresponding to equations (4.19). For \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\rho \in W^{3-1/q}_q(S_R)\), let \(K = K(\mathbf{v}, \rho ) \in H^1_q(B_R)\) be the unique solution of the weak Dirichlet problem:
subject to the boundary condition:
Then, the reduced problem corresponding to problem (4.19) with \(g=\mathbf{g}=\mathbf{h}=0\) is given by the following equations:
Then, for \(\mathbf{f}\in J_q(B_R)\) and \(d \in W^{2-1/q}_q(S_R)\), problems (4.22) and (4.25) are equivalent. In fact, if problem (4.22) admits unique solutions \(\mathbf{v}\in H^2_q(B_R)^N\), \({\mathfrak {p}}\in H^1_q(B_R)\) and \(\rho \in W^{3-1/q}_q(S_R)\), then for any \(\varphi \in \hat{H}^1_{q',0}(B_R)\), we have
because \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). Moreover, from the boundary conditions in (4.22) and (4.24) it follows that
on \(S_R\). The uniqueness of the weak Dirichlet problem leads to \({\mathfrak {p}}= K(\mathbf{v}, \rho )\), and therefore \(\mathbf{v}\) and \(\rho \) satisfy equations (4.25). Conversely, if \(\mathbf{v}\in H^2_q(B_R)^N\) and \(\rho \in W^{3-1/q}_q(S_R)\) satisfy the equations (4.25), then for any \(\varphi \in \hat{H}^1_{q', 0}(B_R)\) we have
Moreover, the boundary conditions of (4.25) and (4.24) gives that
The uniqueness of the weak Dirichlet problem yields that \(\mathrm{div}\,\mathbf{v}=0\), and therefore, \(\mathbf{v}\), \({\mathfrak {p}}= K(\mathbf{v}, \rho )\) and \(\rho \) are solutions of equations (4.22).
Finally, we show the uniqueness of equations (4.21) in the \(L_2\)-framework, which yields Theorem 16. Let \(\mathbf{v}\in H^2_2(B_R)^N\) and \(\rho \in W^{5/2}_2(S_R)\) satisfy the homogeneous equations:
Note that \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). Employing the same argument as in the proof of Theorem 15, we have
In particular, \({\mathcal M}\rho =0\). Multiplying the first equation with \(\mathbf{v}\), integrating the resultant formula on \(B_R\) and using the divergence theorem of Gauss gives that
because \((K(\mathbf{v}, \rho ), \mathrm{div}\,\mathbf{v}) = 0\) as follows from \(\mathrm{div}\,\mathbf{v}=0\) in \(B_R\). From (2.17) it follows that
From the second equation of (4.26) with \({\mathcal M}\rho =0\) it follows that
Combining these formulas yields that
which leads to \(\mathbf{D}(\mathbf{v}) = 0\) and \((\mathbf{v}, \mathbf{p}_k)_{B_R} = 0\) for \(k = 1, \ldots , M\). Thus, we have \(\mathbf{v}=0\). From the first equation of (4.26), we have \(\nabla K(\mathbf{v}, \rho ) = 0\), and so \(K(\mathbf{v}, \rho ) = c\) with some constant c. From the boundary condition of (4.26), we have \(\sigma {\mathcal B}\rho =-c\) on \(B_R\). Integrating this formula on \(S_R\) and using the fact \((\rho , 1)_{S_R}=0\) in (4.27) gives that \(c=0\). Thus, \({\mathcal B}_R\rho =0\) on \(S_R\), but we know (4.27), and so
for some constant \(c > 0\), which shows that \(\rho =0\). This completes the proof of the uniqueness in the \(L_2\) framework, the proof of Theorem 16. \(\square \)
Proof of Theorem 6
We now prove Theorem 6. Let \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) be functions given in Theorem 14 which are solutions of equations (4.6). Notice that \(\psi (ik) = 1\) for \(|k| \ge k_0+4\) and \(\psi (ik) = 0\) for \(|k| \le k_0+3\). For \(k \in {\mathbb Z}\) with \(1\le |k| \le k_0+3\), let
in equations (4.3), and we write solutions \(\mathbf{v}\), \({\mathfrak {q}}\) and \(\eta \) as \(\mathbf{v}_k=\mathbf{v}\), \({\mathfrak {q}}_k={\mathfrak {q}}\) and \(\eta _k = \eta \). Let
and then, \(\mathbf{u}_k\), \({\mathfrak {p}}_k\) and \(\rho _k\) satisfy the equations:
Let \(\mathbf{f}= \mathbf{F}_S\), \(d=D_S\), \(g=G_S\), \(\mathbf{g}= \mathbf{G}_S\) and \(\mathbf{h}= \mathbf{H}_S\) in equations (4.19), and let \(\mathbf{v}\), \({\mathfrak {p}}\) and \(\rho \) be unique solutions of equations (4.19). We write \(\mathbf{u}_S = \mathbf{v}\), \({\mathfrak {p}}_S = {\mathfrak {p}}\) and \(\rho _S = \rho \). Under these preparations, we set
and then \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) are unique solutions of equations (4.1). Moreover, by Theorem 14, Theorem 15, and Theorem 16, we see that \(\mathbf{u}\), \({\mathfrak {p}}\) and \(\rho \) satisfy the estimate (4.2). In fact, for \(f = f_S + \sum _{1\le |k| \le k_0+3} \mathrm{e}^{ikt}f_k + f_\psi \), we have the following estimates:
By Hölder’s inequality, we have
and for any UMD Banach space X, using Lemma 13 and transference theorem, Theorem 9, we have
\(\square \)
4.2 On linearized problem of two-phase problem
In this subsection, we consider the linear equations:
where \(\Omega _+ = B_R\), \(\Omega _- = \Omega {\setminus }(B_R\cup S_R)\), and \({\mathcal M}\), \({\mathcal A}\) and \({\mathcal B}_R\) are the linear operators defined in (2.17). We shall prove the unique existence theorem of \(2\pi \)-periodic solutions of equations (4.29). Our main result in this section is stated as follows.
Theorem 17
Let \(1< p, q < \infty \). Then, for any \(\mathbf{F}_\pm \), D, \(G_\pm \), \(\mathbf{G}_\pm \) and \(\mathbf{H}\) with
problem (4.1) admits unique solutions \(\mathbf{u}_\pm \), \({\mathfrak {p}}_\pm \) and \(\rho \) with
possessing the estimate:
for some constant \(C > 0\).
To prove Theorem 17, the strategy is the same as in the proof of Theorem 6. Therefore, we first consider the \({\mathcal R}\)-solver of the generalized resolvent problem:
for \(k \in {\mathbb R}\). From Theorem 2.1.4 in Shibata and Saito [19] we know the following theorem concerned with the existence of an \({\mathcal R}\)-solver of problem (4.29).
Theorem 18
Let \(1< q < \infty \) and let \({\mathbb R}_{k_0} = {\mathbb R}{\setminus }(-k_0, k_0)\). Let
Then, there exist a constant \(k_0 > 0\) and operator families \({\mathcal A}(ik)\), \({\mathcal P}(ik)\), and \({\mathcal H}(ik)\) with
such that for any \((\mathbf{f}, d, \mathbf{h}, g, \mathbf{g})\) and \(k \in {\mathbb R}_{k_0}\), \(\mathbf{v}= {\mathcal A}(ik){\mathcal F}_k\), \({\mathfrak {q}}= {\mathcal P}(ik){\mathcal F}_k\) and \(\eta = {\mathcal H}(ik){\mathcal F}_k\), where
are unique solutions of equations (4.31), and
for \(\ell =0,1\), \(m=0,1,2\) and \(n=0,1\) with some constant \(r_b\).
Remark 19
-
(1)
Here \(f \in L_q(\dot{\Omega })\) means that \(f_\pm \in L_q(\Omega _\pm )\), and \(f \in H^1_q(\dot{\Omega })\) means that \(f_\pm \in H^1_q(\Omega _\pm )\), and we set
$$\begin{aligned}\Vert f\Vert _{L_q(\dot{\Omega })} = \sum _\pm \Vert f_\pm \Vert _{L_q(\Omega _\pm )}, \quad \Vert f\Vert _{H^1_q(\dot{\Omega })} = \sum _\pm \Vert f_\pm \Vert _{H^1_q(\Omega _\pm )}. \end{aligned}$$Moreover, we define
$$\begin{aligned} \dot{H}^1_q(\dot{\Omega }) = \Big \{\theta \in H^1_q(\dot{\Omega }) \mid \int _{\dot{\Omega }} \theta \,\mathrm{d}x = 0\Big \}. \end{aligned}$$ -
(2)
For f defined on \(\dot{\Omega }\), we set \(f_\pm = f|_{\Omega _\pm }\) and for \(f_\pm \) defined on \(\Omega _\pm \), we set \(f = f_\pm \) on \(\Omega _\pm \). The functions \(F_1\), \(F_2\), \(F_3\), \(F_4\), \(F_5\), \(F_6\), and \(F_7\) are variables corresponding to \(\mathbf{f}\), d, \((ik)^{1/2}\mathbf{h}\), \(\mathbf{h}\), \((ik)^{1/2}g\), g, and \(ik\, \mathbf{g}\), respectively.
-
(3)
We define the norm \(\Vert \cdot \Vert _{{\mathcal X}_q(\Omega )}\) by setting
$$\begin{aligned} \Vert (F_1, \ldots , F_7)\Vert _{{\mathcal X}_q(\Omega )}= & {} \Vert (F_1, F_5, F_7)\Vert _{L_q(\dot{\Omega })} + \Vert F_2\Vert _{W^{2-1/q}_q(S_R)} + \Vert F_6\Vert _{H^1_q(\dot{\Omega })} \\&+ \Vert F_3\Vert _{L_q(\Omega )} + \Vert F_4\Vert _{H^1_q(\Omega ))}. \end{aligned}$$
Let \(\varphi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+2}\) and zero for \(k \not \in {\mathbb R}_{k_0+1}\), and let \(\psi (ik)\) be a function in \(C^\infty ({\mathbb R})\) which equals one for \(k \in {\mathbb R}_{k_0+4}\) and zero for \(k \not \in {\mathbb R}_{k_0+3}\). For \(f \in \{\mathbf{F}_\pm , G_\pm , \mathbf{G}_\pm , D, \mathbf{H}\}\), we set
We consider the high frequency part of the equations (4.29):
By Theorem 8, Theorem 9, and the analogue of (4.5) resulting from (4.35), we have immediately the following theorem.
Theorem 20
Let \(1< p, q < \infty \). Then, for any functions \(\mathbf{F}\), G, \(\mathbf{G}\), D, and \(\mathbf{H}\) with
We let
where we have set
Then, \(\mathbf{u}_\psi \), \({\mathfrak {p}}_\psi \) and \(\rho _\psi \) are the unique solutions of equations (4.33), which possess the following estimate:
for some constant \(C > 0\). Here, we have set
We now consider the lower frequency part of solutions of equations (4.29). Namely, we consider equations (4.31) for \(k \in {\mathbb R}\) with \(1 \le |k| < k_0+4\). We shall show the following theorem.
Theorem 21
Let \(1< q < \infty \) and \(k \in {\mathbb Z}\) with \(|k| \le k_0+3\). Then, for any \(\mathbf{f}_\pm \in L_q(\Omega _\pm )^N\), \(g_\pm \in H^1_q(\Omega _\pm )\), \(d \in W^{2-1/q}_q(S_R)\), \(\mathbf{h}\in H^1_q(\Omega )^N\), and \(\mathbf{g}_\pm \in L_q(\Omega _\pm )^N\), problem (4.31) admits unique solutions \(\mathbf{v}_\pm \in H^2_q(\Omega _\pm )^N\), \({\mathfrak {q}}_\pm \in H^1_q(\Omega _\pm )\) with \(\int _\Omega {\mathfrak {q}}\,\mathrm{d}x=0\), and \(\eta \in W^{3-1/q}_q(S_R)\) possessing the estimate:
for some constant \(C > 0\).
Proof
The strategy of the proof is the same as that in Theorem 15. The only difference is the reduced problem. First, we can reduce equations (4.31) to equations:
For any \(\mathbf{v}_\pm \in H^2_q(\Omega _\pm )^N\) and \(\rho \in W^{3-1/q}_q(S_R)\), let \(K=K(\mathbf{v}, \rho ) \in \dot{H}^1_q(\dot{\Omega })\) be the unique solution of the weak Neumann problem:
subject to the transmission condition:
where \(\mu \) is piecewise constant defined by \(\mu |_{\Omega _\pm } = \mu _\pm \). Here and in the following, \(\dot{H}^1_q(\Omega )\) is defined by setting
The reduced problem corresponding to equations (4.35) is
Let \(J_q(\dot{\Omega })\) be the solenoidal space defined by setting
For any \(\mathbf{f}\in J_q(\dot{\Omega })\) and \(d \in W^{2-1/q}_q(S_R)\), problems (4.35) and (4.38) are equivalent. In fact, if problem (4.35) admits unique solutions \(\mathbf{v}\in H^2_q(\dot{\Omega })^N\), \({\mathfrak {p}}\in \dot{H}^1_q(\dot{\Omega })\) and \(\rho \in W^{3-1/q}_q(S_R)\), then using the divergence theorem of Gauss and noting that \([[\varphi ]]=0\) on \(S_R\) gives that for any \(\varphi \in \dot{H}^1_{q'}(\Omega )\),
because \(\mathrm{div}\,\mathbf{v}=0\) on \(\dot{\Omega }\). Moreover, the transmission conditions in (4.35) and (4.37) gives that
Thus, the uniqueness of the weak Neumann problem in \(\dot{H}^1_q(\dot{\Omega })\) yields that \({\mathfrak {p}}- K(\mathbf{v}, \rho ) = 0\) in \(\Omega \). Thus, \(\mathbf{v}\) and \(\rho \) satisfy the equations (4.38).
Conversely, if \(\mathbf{v}\in H^2_q(\dot{\Omega })^N\) and \(\rho \in W^{3-1/q}_q(S_R)\) satisfy equations (4.38), then the divergence theorem of Gauss gives that for any \(\varphi \in \dot{H}^1_{q'}(\Omega )\) we have
Moreover, the transmission conditions in (4.38) and (4.37) give that
Thus, the uniqueness of this weak Neumann problem yields that \(\mathrm{div}\,\mathbf{v}=c\) in \(\dot{\Omega }\) for some global constant c. Now the divergence theorem of Gauss and the boundary conditions in (4.38) yield \(c=0\), that is, \(\mathrm{div}\,\mathbf{v}=0\), which shows that \(\mathbf{v}\), \({\mathfrak {p}}= K(\mathbf{v}, \rho )\) and \(\rho \) satisfy equations (4.35).
Employing the same argument as that in the proof of Theorem 15, we see that to prove Theorem 21, it is sufficient to prove the uniqueness of solutions to equations (4.38) in the \(L_2\) framework. Thus, we choose \(\mathbf{v}\in H^2_2(\dot{\Omega })^N\) and \(\rho \in W^{5/2}_2(S_R)\) be solutions of the homogeneous equations:
and we shall show that \(\mathbf{v}=0\) and \(\rho =0\). Notice that \(\mathrm{div}\,\mathbf{v}= 0\) on \(\dot{\Omega }\). Moreover, by \([[\mathbf{v}]]=0\), we have \(\mathbf{v}\in H^1_q(\Omega ) \cap H^2_q(\dot{\Omega })\). Integrating the second equation in (4.39) over \(S_R\) and using the divergence theorem of Gauss on \(\Omega _+ = B_R\) gives that
because \(\mathrm{div}\,\mathbf{v}_+ = 0\) on \(B_R\), and so \((\rho , 1)_{S_R}=0\). Moreover, multiplying the second equation in (4.39) by \(x_j\) and integrating over \(S_R\), similar arguments lead to
because \((1, x_j)_{S_R}=0\), and \((x_k, x_j)_{S_R} = 0\) for \(j\not =k\). Since \((x_j, x_j)_{S_R} = (R^2/N)|S_R| > 0\), we have \((\rho , x_j) = 0\). Summing up, we have proved
In particular, \({\mathcal M}\rho =0\).
We now prove that \(\mathbf{v}= 0\). Multiplying the first equation of (4.39) with \(\mathbf{v}\) and integrating the resultant formula over \(\dot{\Omega }\) and using the divergence theorem of Gauss gives that
because \(\mathrm{div}\,\mathbf{v}= 0\) in \(\dot{\Omega }\). By the second equation of (4.39) with \({\mathcal M}\rho =0\), we have
where we have used \(\mathbf{n}=R^{-1}x= R^{-1}(x_1,\ldots , x_N)\) for \(x \in S_R\). This also yields
Moreover, since \(\rho \) satisfies (4.40), we know that
for some positive constant c, and therefore we have \(\mathbf{D}(\mathbf{v})=0\). Since \(\mathbf{v}\in H^1_q(\Omega )\) and \(\mathbf{v}=0\) on \(S_-\), we have \(\mathbf{v}=0\).
Finally, the first equation of (4.39) yields that \(\nabla K(\mathbf{v}, \rho ) = 0\), which shows that \(K(\mathbf{v}, \rho )\) is constant in \(\dot{\Omega }\). Thus, \([[K(\mathbf{v}, \rho )]]\) is constant. Integrating the third equation of (4.39) yields that
where we have used (4.40). In particular, \(K(\mathbf{v}, \rho )\) is a constant globally in \(\Omega \). Finally, we have \({\mathcal B}_R\rho = 0\) on \(S_R\), which, combined with (4.40) leads to \(\rho =0\). This completes the proof of uniqueness for equations (4.38) in the \(L_2\) framework. Therefore, we have proved Theorem 21. \(\square \)
Proof of Theorem 17
Employing the same argument as in the proof of Theorem 6 and using Theorem 20 and Theorem 21, we can prove Theorem 17. We may omit the detailed proof. \(\square \)
5 Proofs of main results
In this section, we shall prove Theorem 4. The proof of Theorem 5 is parallel to that of Theorem 4, and so we may omit it. We prove Theorem 4 with the help of the usual Banach fixed-point argument, and we define an underlying space \({\mathcal I}_\epsilon \) with some small constant \(\epsilon > 0\) determined later by setting
where we have set
In view of (2.9), we define \(\xi (t)\) by setting
where c is a constant for which
We choose \(\delta > 0\) so small that the map \(x =\Phi (y, t)= y + \Psi (y, t)\) with \(\Psi (y, t)=\Psi _h(y, t) = R^{-1}H_h(y, t)y + \xi (t)\) is one to one. In particular, we may assume that \(\delta > 0\) and the inverse map: \(y = \Xi (y, t)\) is well-defined and has the same regularity property as \(\Phi (y, t)\). In particular, we may assume that
Since \(\epsilon > 0\) will be chosen small eventually, we may assume that \(0< \epsilon < 1\), and so for example, we estimate \(\epsilon ^2 < \epsilon \) if necessary. Let \((\mathbf{v}, h) \in {\mathcal I}_\epsilon \) and let \(\mathbf{u}\) and \(\rho \) be solutions of linearized equations:
In view of Theorem 6, we shall show that
for some constant \(C > 0\) independent of \(\epsilon > 0\). In the following, C denotes generic constants independent of \(\epsilon > 0\), the value of which may change from line to line. Before starting with the estimates of the nonlinear terms, we summarize some inequalities which are useful for our estimations. The following inequalities follow from Sobolev’s inequality and the estimate of the boundary trace:
for \(N< q < \infty \) with some constant C. The following inequalities follow from real interpolation theorem and the periodicity of functions, which will be used to estimate the \(L_\infty \) norm with respect to the time variable of lower order regularity terms with respect to the space variable x:
In fact, to obtain (5.8) we use the following well-known result: Let X and Y be two Banach spaces such that Y is continuously embedded into X, and then \(C([0, \infty ), (X, Y)_{1-1/p,p})\) is continuously embedded into \(H^1_p((0, \infty ), X) \cap L_p((0, \infty ), Y)\) and
For its proof, we refer to [9, 21].
We start with the estimate of \(\mathbf{F}(\mathbf{v}, h)\). From (3.11), we have
which, combined with (5.8) and (5.1), leads to
because \(1 < 2(1-1/p)\) and \(2-1/q < 3-1/p-1/q\). From (3.12), it follows that
To estimate \(\mathbf{F}_2(\mathbf{v}, h)\), we recall
and that \(\Psi (y, t) = R^{-1}H_h(y, t)y + \xi (t)\), where \(\xi (t)\) is given by
By (5.7), (2.5), (5.8), the fact that \(2-1/q < 3-1/p-1/q\), and (5.1), we have
From (5.10) and (5.1), it follows that
In particular, by (5.11) and (5.13), we have
Combining (5.1) and (5.14) gives that
which, combined with (5.9), leads to
We next estimate \(\tilde{d}(\mathbf{v}, h)\). By (3.25) and (5.1),
Since we assume that \(2/p + N/q < 1\), we can choose \(\kappa > 0\) so small that \(2+N/q + \kappa -1/q < 3-1/p-1/q\) and \(1 + N/q + \kappa < 2(1-1/p)\), and then by Sobolev’s inequality and (5.8) we have
where we have used (2.5) in the last inequality. Then, in particular, using again (2.5), we have
Thus, applying (5.12) to the formula in (2.11) and using (5.1) and (5.7) gives that
On the other hand, by (5.11),
and so
which, combined with (5.18), leads to
We next consider \(\mathbf{g}(\mathbf{v}, h)\) given in (3.6), where \(\rho \) is replaced by h. We may write
where \(\mathbf{k}\) denotes variables corresponding to \((H_h, \nabla H_h)\) and \(\mathbf{V}_\mathbf{g}\) is a \(C^\infty \) function defined on \(|\mathbf{k}| < \delta \). We write
and so, by (5.11), (2.5), we have
We next estimate \(g(\mathbf{v}, h)\) and \(\mathbf{h}(\mathbf{v}, h)=(\mathbf{h}'(\mathbf{v},h),h_N(\mathbf{v},h))\) given in (3.6), (3.31) and (3.34), where \(\rho \) is replaced by h. We may write
where \(\mathbf{k}\) are variables corresponding to \((H_h, \nabla H_h)\) and \(V_g(\mathbf{k})\) is some matrix of \(C^\infty \) functions defined on \(|\mathbf{k}| < \delta \). To estimate g, we use the following two lemmas.
Lemma 22
Let \(1<p < \infty \) and \(N< q < \infty \). Let
Then, we have
for some constant \(C > 0\).
Proof
By (5.7), we have
To estimate the \(H^{1/2}\) norm, we use the complex interpolation relation:
where \((\cdot , \cdot )_{1/2}\) denotes a complex interpolation of order 1/2. By (5.7), we have
Thus, by (5.23), we have
Since \(\Vert g\Vert _{L_p((0, 2\pi ), H^{1/2}_q(B_R))} \le C\Vert g\Vert _{L_p((0, 2\pi ), H^1_q(B_R))}\), combining (5.22) and (5.24) leads to (5.21), which completes the proof of Lemma 22. \(\square \)
Lemma 23
Let \(1< p, q < \infty \). Then, there exists a constant C such that for any u with
we have
for some constant \(C > 0\).
Proof
As was proved in the proof of Proposition 1 in Shibata [17], there exist two operators \(\Phi _1\) and \(\Phi _2\) with
such that for any \(g \in H^2_q(B_R)\), we have
and
for \(\ell =0,1\) with some constant \(r_b\). Thus, by Weis’ operator-valued Fourier multiplier theorem, Theorem 8, and transference theorem, Theorem 9, we have (5.25), which completes the proof of Lemma 23. \(\square \)
By (5.1), (2.5), (5.7) and (5.17), we have
Thus, by Lemma 22, Lemma 23, and (5.1), we have
Analogously, recalling the definition of \(\mathbf{h}(\mathbf{v}, h)=(\mathbf{h}'(\mathbf{v},h),h_N(\mathbf{v},h))\) given in (3.31) and (3.34), where \(\rho \) is replaced by h, and using Lemma 22, we have
Since \(H^{1/2}_p((0, 2\pi ), L_q(B_R)) \supset H^1_p((0, 2\pi ), L_q(B_R))\), we have
and so using Lemma 23, (2.5), and (5.1), we have
Combining (5.15), (5.19), (5.20), (5.26), and (5.27) gives (5.6). Applying Theorem 6 to equations (5.5) and using (5.6) and (5.16) gives that
for some constants \(M_1\) and \(M_2\) independent of \(\epsilon \in (0, 1)\). Finally, we estimate \(\Vert \partial _t\rho \Vert _{L_\infty ((0, 2\pi ), W^{1-1/q}_q(S_R))}\). From the third equation in equations (5.5), we have
Therefore, by (5.1), (5.7), (5.8), (5.11), (5.12), and (5.13), we have
which, combined with (5.28), leads to
for some constants \(M_1'\) and \(M_2'\) independent of \(\epsilon \in (0, 1)\). We choose \(\epsilon > 0\) so small that \(M'_2\epsilon < 1/2\) and we assume that \(M_1'\Vert \mathbf{f}\Vert _{L_p((0, 2\pi ), L_q(D))} \le \epsilon /2\). Then, we have
Moreover, by (2.5) and (5.8), we have
Choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(0< M_3\epsilon < \delta \), and so \((\mathbf{u}, \rho ) \in {\mathcal I}_\epsilon \). If we define a map \(\Phi \) acting on \((\mathbf{v}, h) \in {\mathcal I}_\epsilon \) by setting \(\Phi (\mathbf{v}, h) = (\mathbf{u}, \rho )\), and then \(\Phi \) is a map from \({\mathcal I}_\epsilon \) into itself. Employing a similar argument as for proving (5.30), we see that for any \((\mathbf{v}_i, h_i) \in {\mathcal I}_\epsilon \) (\(i = 1,2\)),
Choosing \(\epsilon > 0\) smaller if necessary, we may assume that \(M_4\epsilon \le 1/2\), and so \(\Phi \) is a contraction map on \({\mathcal I}_\epsilon \). The Banach fixed-point theorem yields the unique existence of a fixed point \((\mathbf{v}, \rho ) \in {\mathcal I}_\epsilon \) of the map \(\Phi \), that is \((\mathbf{v}, \rho ) = \Phi (\mathbf{v}, \rho )\), which is the required solution of equations (2.16). This completes the proof of Theorem 4.
Notes
For any f(x, t) defined on \(\Omega _t\), we have
$$\begin{aligned} \frac{d}{\mathrm{d}t}\int _{\Omega _t}f(x, t)\,\mathrm{d}x = \int _{\Omega _t} (\partial _tf + \mathrm{div}\,(f\mathbf{u}))\,\mathrm{d}x, \end{aligned}$$which is called the Reynolds transport theorem.
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Mads Kyed: Joint Appointment Professor of Waseda University. Y. Shibata: Partially supported by JSPS Grant-in-aid for Scientific Research (A) 17H0109, and Top Global University Project. Adjunct Faculty member in the Department of Mechanical Engineering and Materials Science, University of Pittsburgh.
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Eiter, T., Kyed, M. & Shibata, Y. On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations. J. Evol. Equ. 21, 2955–3014 (2021). https://doi.org/10.1007/s00028-020-00619-5
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DOI: https://doi.org/10.1007/s00028-020-00619-5