Nonstationary Stokes system in Sobolev–Slobodetski spaces

We consider an initial-boundary value problem for the nonstationary Stokes system in a bounded domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb R ^3$$\end{document} with slip boundary conditions. We prove the existence in the Hilbert–Sobolev–Slobodetski spaces with fractional derivatives. The proof is divided into two main steps. In the first step by applying the compatibility conditions an extension of initial data transforms the considered problem to a problem with vanishing initial data such that the right-hand sides data functions can be extended by zero on the negative half-axis of time in the above mentioned spaces. The problem with vanishing initial data is transformed to a functional equation by applying an appropriate partition of unity. The existence of solutions of the equation is proved by a fixed point theorem. We prove the existence of such solutions that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in H^{l+2,l/2+1}(\Omega \times (0,T)),\,\nabla p\in H^{l,l/2}(\Omega \times (0,T)),\,v$$\end{document}—velocity, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}—pressure, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l\in \mathbb R _+\cup \{0\},\,l \ne [l]+\frac{1}{2}$$\end{document} and the spaces are introduced by Slobodetski and used extensively by Lions–Magenes. We should underline that to show solvability of the Stokes system we need only solvability of the heat and the Poisson equations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R ^3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R _+^3$$\end{document}. This is possible because the slip boundary conditions are considered.


Introduction
The aim of this paper is solvability in Sobolev-Slobodetski spaces of the Hilbert type of the following slip boundary value problem for the nonstationary Stokes system , v 3 (x, t)) ∈ R 3 is the velocity of the fluid, x = (x 1 , x 2 , x 3 ) the Cartesian system in R 3 , f 0 = ( f 01 (x, t), f 02 (x, t), f 03 (x, t)) ∈ R 3 the external force field, p = p(x, t) ∈ R the pressure. Moreover,n is the unit outward vector normal to S,τ α , α = 1, 2, tangent to S, γ ≥ 0 is the constant slip coefficient.
By T(v, p) we denote the stress tensor of the form where ν > 0 is the constant viscosity coefficient, I is the unit matrix and D(v) is the dilatation tensor of the form To formulate the main result of this paper we need. Definition 1.1 (see equivalent Definition 2.1) By W l,l/2 2 (Q T ), l ∈ R + ∪ {0}, Q ⊂ R n , we define the anisotropic Sobolev-Slobodetski space as a set of functions with the following finite norm In the case of integer l the middle integral and for l even the last two integrals disappear.
To simplify notation we introduce H l,l/2 (Q T ) = W l,l/2 2 (Q T ). Let Q be a submanifold of R n . Then norm (1.3) is defined by an appropriate partition of unity.

6)
∂ τ α is the tangent derivative to S directed along the tangent vectorτ α , α = 1, 2 A very important step in the existence theory of solutions to the nonstationary problem (1.1) is its transformation to a problem with time t ∈ R. For this purpose we transform problem (1.1) to a problem with vanishing initial data by its appropriate extension. Next such compatibility conditions on the r.h.s. functions in Eq. (1.1) 1,2,3,4 must be assumed that after the above extension of the initial data they can be extended by zero for t < 0 in the considered spaces.
For this we extend the initial data in such a way thatṽ 0 | t=0 = v 0 ,ṽ 0 ∈ H 2+s,1+s/2 ( T ) and (1.9) Then we impose the following compatibility conditions Definition 1. 3 Assume thatṽ, f 0 , g 0 , h 0 satisfy the relations (1.10) The main result of this paper reads For c 0 independent of T see Sect. 6.

Remark 1.5
To describe the compatibility conditions (1.10) we have first to construct functionṽ 0 . From (1.10) the following iterative relation follows (1.14) Calculating ∂ k tṽ 0 | t=0 from (1.12) and (1.13) step by step we obtain where b (k) is a differential operator with respect to space variables of order 2k for v 0 and 2k − 2 for f 0 | t=0 , and k is described above.  (1.16) where b 0 = v 0 . Havingṽ 0 calculated above conditions (1.10) 2,3,4 are restrictions on g 0 and h 0 . In this paper we prove the existence of solutions to problem (1.1) in the Hilbert-Sobolev-Slobodetski spaces H 2+s,1+s/2 ( T ), where s ∈ R + ∪ {0}. The proof is divided into two main steps: existence of solutions to problem (1.1) vanishing at time t ≤ 0 (see Sect. 4) and appropriate extension of the initial data which transforms problem (1.1) to the problem (3.1) with vanishing initial data (see Sect. 5). The first step is much more difficult and crucial in the proof. It basis on local considerations (see Sect. 3) and some ideas of the regularizer (see [10,Ch. 4]).
The local considerations are connected with localization of problem (3.1) by an appropriate partition of unity (see Sect. 2). We distinguish two kinds of subdomains: interior and close to the boundary.
Then applying the partition of unity a solution to problem (3.1) is calculated in the form (4.1), where v (k) and p (k) are expressed explicitly (see Sect. 3). Problem (3.1) is considered in spaces H σ,σ/2 γ , γ ≥ 0, σ > 0 (see Definition 2.1), which are invariant with respect to an extension by zero for t ≤ 0. The spaces are appropriate for examining the solvability of parabolic problems and were introduced by Agranovich and Vishik (see [1]) and were strongly developed by Solonnikov in [12] (see also [9,Vol. 2]).
The most natural way to find estimates and solve nonstationary parabolic problems with vanishing initial data is to find the Fourier-Laplace transforms of solutions and estimate them in spacesH The main goal of this paper is to prove the existence of solutions to problem (3.1) using only the existence of solutions to the heat equations in R 3 × R, R 3 + × R and to the Poisson equation in R 3 , R 3 + . This can be made by applying the Fourier-Laplace transform and use estimates in spacesH σ,σ/2 γ andH σ γ respectively. This is possible because the Helmholtz-Weyl decomposition reduces solvability of (3.1) to solvability of the heat equation and the Poisson equation in the above mentioned spaces.
To show that our method is simple we recall the classical approach of solvability of an initial-boundary value problem for the nonstationary Stokes system (see [12,13,15,16]). After appropriate transformation the nonstationary Stokes system with vanishing initial data in the half space R 3 + × R, where R 3 + = {x ∈ R 3 : x 3 > 0} is derived. Then after the Laplace transform with respect to t and the Fourier transforms with respect to variables x 1 , x 2 tangent to the plane x 3 = 0 a system of ordinary differential equations with respect to x 3 is formulated. Solving this system (see [12] for the Neumann boundary conditions and [7] for some parabolic system with slip boundary condition) and applying the definition ofH σ,σ/2 γ the appropriate estimate follows.
In the L p -approach we can distinguish two ways. First: solving the system in the half-space we construct a solution by using a corresponding Green function and next it is estimated by the Calderon-Zygmund theorem. Second: solving the ordinary differential system for the Fourier-Laplace transform we apply the Marcinkiewicz-Mikhlin theorem and obtain the estimate directly. The first approach was employed by Solonnikov in [13][14][15][16] in the case of non-slip and the Neumann boundary conditions. The second approach for the Neumann and the slip boundary conditions was used in [2,3,11] . In the fourth-coming paper we are going to extend the presented in this paper proof to the L p -case.

Notation and auxiliary results
First we introduce the partition connected with a partition of unity {ω (k) , (k) Hence ω (k) ⊂ (k) and k∈M∪N ω (k) = k∈M∪N (k) = , ω (k) , (k) for k ∈ M are interior subdomains but ω (k) , (k) , k ∈ N are neighbourhoods near S. Moreover, we introduce functions ϑ (k) We assume that at most N 0 of (k) has nonempty intersection, so There is no relation between N 0 and λ. Hence we have By ξ (k) we denote an interior point of ω (k) and (k) for k ∈ M and the center of To consider problem (1.1) in a neighbourhood of S we have to make it locally flat. Since problem (1.1) is invariant with respect to a translation and a rotation we can introduce a local system of coordinates with origin in ξ (k) , k ∈ N . We shall denote it by y = (y 1 , y 2 , y 3 ) and y = Y k (x), where Y k is a composition of a translation and a rotation, which is such that the part S (k) = S ∩¯ (k) of the boundary is described by . Then we introduce new coordinates and denote the mapping by z = k (y). , We assume that the setsω (k) ,ˆ (k) are described in the local coordinates at ξ (k) by the inequalitites Finally, we introduce the notation for k ∈ N and in the case of vector-valued function u we haveû 3 is an orthogonal matrix implied by the transformation y i = 0 i j Y i + r i and r i describes translation. Moreover, Definition 2. 1 We use the anisotropic Sobolev-Slobodetskii spaces W l,l/2 2 where Q is either or S, with the norm (see [1,7,12]) for integer l, and is the integer part of l. For Q = S the above norm is introduced by using local charts and a partition of unity. Finally, for integer l/2, and for l/2 noninteger. We shall use the simplified notation To consider problems with vanishing initial conditions we need a space of functions which admits a zero extension to t < 0. Therefore, for every γ ≥ 0, we introduce the space H l,l/2 γ (Q T ) with the norm (see [1,7,12]) In accordance with [1] Let us introduce the Fourier-Laplace transform for functions defined in and for functions defined in R 3 the Laplace transform is defined for Res ≥ γ . By the Paley-Wiener theorem the Laplace transformũ(x, s) is a holomorphic function of s for Res > γ .

Definition 2.3 ByH
By the properties of the Parseval identity and Lemma 2.1 in [12] we have Lemma 2.4 For any γ ≥ 0 there exist constants c 1 and c 2 independent of u and γ, l such that Proof The proof is almost the same as the corresponding proof in [12,Sect. 2]. Since the result is very important we recall it for the reader convenience.
By the Fourier-Laplace transform (2.3) and the Parseval identity we have The above identity implies Moreover, for noninteger l/2 we have where k = l 2 .
Since the expression behaves as τ 2 near τ = 0 the above considerations imply (2.9) for γ = 0. For l/2 integer the above results also holds. This concludes the proof. We need the following interpolation inequality Lemma 2.6 (see [12,Sect. 4 For any finite T the norms of spaces H l,l/2 The proof of the lemma follows directly from the definition of H l,l/2 γ and the fact that elements of that space vanish for t < 0.
Finally we consider the problems and Proof Let us restrict our considerations to problem (2.15). Extending f on x n < 0 by the Hestenes-Whitney method we solve the equation Applying transform (2.4) we get where τ = s + |ξ | 2 . Solving (2.19) yieldsũ =de −τ x n . Using spacesH l,l/2 γ (R n + × R) introduced in Definition 2.3 and then Remark 2.5 (for details see [7,12]) we prove the lemma.
Let us consider the elliptic problem (2.21)

Local considerations
The aim of this section is to transform problem (1.1) with vanishing initial data to the Poisson equation in R 3 , R 3 + and the heat equation in R 3 × R, R 3 + × R for localized velocity and pressure. The localization is made by an appropriate partition of unity and the Helmholtz-Weyl decomposition.
Therefore we consider the following problem where we assumed that functions f, g, h α , α = 1, 2, 3, are extended by zero for t < 0 and next by the Hestenes-Whitney method for t > T .
First we consider problem (3.1) in an interior subdomain. Multiplying (3.1) by ζ (k) , k ∈ M, and using notation (2.2) we obtain v (k) To apply the Helmholtz-Weyl decomposition we have to work with divergence free functions. For this purpose we introduce a function ϕ (k) by Defining the new function we transform problem (3.2) into the following problem 3 such that and η (k) is a solution to the problem Then problem (3.5) splits up into where η (k) is a solution to (3.8) and Now we consider a neighbourhood near the boundary. Then for k ∈ N instead of (3 (3.11) where f (k) 1 and g (k) 1 have the same form as in the case k ∈ M and the summation convention over the repeated indices is assumed.
Next we apply the mapping k and introduce the notation and χ k = y 3 − F k (y 1 , y 2 ). Applying mapping k and notation (2.1) problem (3.11) takes the form v (k) Let us express problem (3.12) (3.13) Let ϕ (k) be a solution to the Neumann problem for the Poisson equation (3.14) Introducing the new function (3.15) we see that (u (k) ,p (k) ) is a solution to the problem We need the Helmholtz-Weyl decompositioñ Then (3.18) Using the decomposition (3.17) in (3.16) yields (3.19) and u (k) is a solution to the problem Hence to prove the existence of solutions to (3.1) some fixed point argument must be applied. To make this possible the pressure p occuring in the r.h.s. of Eqs. (3.8), (3.10), (3.18), (3.20) must be expressed in terms of v. This is possible for slip boundary conditions because in this case p can be calculated from the Neumann problem to the Poisson equations (see [18]). The existence of solutions to (3.1) is proved in Sect. 4.

Existence of solutions with vanishing initial data
In this section we prove the existence of solutions to problem (3.1). To show this we use the local considerations from Sect. 3. By the property of the partition of unity we have where where E(x − y) is the fundamental solution to the Laplace equation and (4.4) Next where p, f, g, h)), k ∈ M (4.7) where L (k) is an operator determined by relations (4.2)-(4.6). Let k ∈ N . Let Z k be such operator that for any function space X . Then where v (k) andṽ (k) are solutions to problems (3.11) and (3.12), respectively. In view of (3.15) we haveṽ where u (k) is a solution to (3.16) and ϕ (k) is the function described by (3.14). Finally, we obtainp where η (k) is a solution to (3.18). Then From (4.8)-(4.11) we obtain Relations (4.1), (4.7), (4.12) imply p, f, g, h).  L(v, p, f, g, h). Then solvability of (4.13) means an existence of a fixed point of the mapping (4.14) For this we need to show that is a contraction. This, however, is impossible if p appears in (4.13). Hence to show the contraction we need a global estimate for pressure on . Let ϕ be a solution to the Neumann problem ϕ = g in , n · ∇ϕ = h 3 on S, (4.15) ϕdx = 0, (4.16) with the following compatibility condition Introducing the new functions Then q is a solution to the problem (4.20) From [18] we havē n · u| S = a αβ u α,τ β + a α u α + b α u α,n − d α,τ α , (4.21) where the summation convention with respect to α, β is assumed, α, β = 1, 2, τ α , α = 1, 2, are tangent coordinates to S, n is the normal coordinate to S, a αβ , a α , b α ∈ C s if S ∈ C s+2 .
Proof Problem (4.20) is elliptic with respect to spatial variables where time t is a parameter. Therefore to obtain an estimate for it is sufficient to obtain estimates for In view of the Laplace transform appeared in (2.3)  Multiplying (4.20) 1 by q and integrating over we obtain In view of (4.20) 2 and application of the Hölder and the Young inequalities to the last term on the r.h.s. we obtain the inequality In view of (4.21) we have where in the second inequality we used (4.18) 1 and (4.19) 4 .
By the Poincare inequality and (4.20) 3 we have Using the above estimates in (4.23) yields Applying the Laplace transform to (4.20) and (4.21), multiplying the results by |γ + iξ 0 | s/2 , taking the inverse Laplace transform and repeating the considerations leading to (4.24) we obtain For solutions to (4.20) we have  [9,Vol. 1]. The proof can be also made directly using [7,12]. Applying the partition of unity problem (4.20) is transformed to the local problems Applying the Fourier transform with respect to x to the problem in R 3 + we get Solving we getũ =be −|ξ |x 3 . Using the definition of space H l (R 3 + ), Proof Time dependence is not essential in proofs of (4.28) and (4.29) because (4.15), (4.16) is an elliptic problem and time is only a parameter. Since s is real we prove the existence and estimates (4.28), (4.29) using [9, Ch. 2, Sect. 5.4]. The proofs can be made as follows. By a partition of unity we transform problem (4.15) to local problems in R 3 and in R 3 + . Then estimates in spaces H s , s ∈ R + , follow by interpolation having estimates in H k , k ∈ N. For a bounded domain we can apply considerations from [9,Ch. 1,Sect. 7,9]. This concludes the proof.
where E is the fundamental solution to the Laplace equation.
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