Mixed Initial-Boundary Value Problem for the Three-Dimensional Navier-Stokes Equations in Polyhedral Domains

We study a mixed initial-boundary value problem for the Navier-Stokes equations, where the Dirichlet, Neumann and slip boundary conditions are prescribed on the faces of a three-dimensional polyhedral domain. We prove the existence, uniqueness and smoothness of the solution on a time interval $(0,T^*)$, where $0


The domain.
It is well known that the regularity results for solutions of elliptic problems in domains with edges or with the mixed boundary conditions are closely related to the properties of the boundary of the domain. Hence we specify several attributes of the domain Ω, which will be used later. We assume that (i) Γ i (the faces of Ω), i = 1, . . . , n, are open two-dimensional manifolds of class C ∞ ; (ii) the boundary ∂Ω consists of smooth faces Γ i (defined above) and smooth (of class C ∞ ) nonintersecting curves M k (the edges), k = 1, . . . , m, vertices on ∂Ω are excluded; (iii) for every A ∈ M k , k = 1, . . . , m, there exists a neighborhood U A and a diffeomorphic mapping ω A > 0 denotes the angle at the edge M k , A ∈ M k , and B A is the unit ball (r, ϕ denote the polar coordinates in the (x 1 , x 2 )-plane); (iv) Γ i ∈ Γ D , i.e. i ∈ J 1 , forms at least one of the adjoining faces of every edge M k , k = 1, . . . , m; (v) 1.3. Basic notation and some function spaces. Vector functions and operators acting on vector functions are denoted by boldface letters. Unless specified otherwise, we use Einstein's summation convention for indices running from 1 to 3.
Let E := u ∈ C ∞ (Ω) 3 ; div u = 0, supp u ∩ Γ D = ∅, supp u · n ∩ Γ G = ∅ and V k,p be a closure of E in the norm of W k,p (Ω) 3 , k ≥ 0 (k need not be an integer) and 1 ≤ p < ∞. Then V k,p is a Banach space with the norm of the space W k,p (Ω) 3 . For simplicity, we denote V 1,2 and V 0,2 , respectively, as V and H. Note, that V and H, respectively, are Hilbert spaces with scalar products and they are closed subspaces of spaces W 1,2 (Ω) 3 and L 2 (Ω) 3 , respectively. In 4 e ij (u) denotes the matrix with the components e ij (u) = 1 2 Further, define the space where u and f are corresponding functions via 5.
Let u, v, w ∈ W 1,2 (Ω) 3 . We will use the notation Throughout the paper, we will always use positive constants c, c 1 , c 2 , . . . , which are not specified and which may differ from line to line.

Formulation of the problem
Let T ∈ (0, ∞), Q = Ω × (0, T ). The classical formulation of our problem is as follows: Here f is a body force and u 0 describes an initial velocity. We assume that functions u, P, f and u 0 are smooth enough and the compatibility conditions u 0 = 0 on Γ D and u 0 · n = 0 on Γ G hold. For simplicity we suppose that ν = 1 throughout the paper.
We can formulate our problem: for every v ∈ V and for almost every t ∈ (0, T ) and The main difficulties of problem 12-13 consist in the fact that, because of the artificial boundary condition 10, we cannot prove that b(u, u, u) = 0. Consequently, we are not able to show that the kinetic energy of the fluid is controlled by the data of the problem and the solutions of 12-13 need not satisfy the energy inequality. This is due to the fact that some uncontrolled "backward flow" can take place at the open parts Γ N of the domain Ω and one is not able to prove global (in time) existence results. In [5]- [7], Kračmar and Neustupa prescribed an additional condition on the output (which bounds the kinetic energy of the backward flow) and formulated steady and evolutionary Navier-Stokes problems by means of appropriate variational inequalities. In [11], Kučera and Skalák prove the local-in-time existence and uniqueness of a "weak" solution of the non-steady Navier-Stokes problem with boundary condition 10 on the part of the boundary ∂Ω, such that (14) u The requirement u 0 ∈ D represents an implicit compatibility condition imposed on the initial data.
under some smoothness restrictions on u 0 and P. In [10], Kučera supposed that the problem is solvable in suitable function class with some given data (the initial velocity and the right hand side). The author proved that there exists a unique solution for data which are small perturbations of the previous ones.
In [3], Beneš and Kučera proved local existence of solutions to the Navier-Stokes system with the so called "do nothing" boundary condition for sufficiently smooth data in a stronger (spatially) sense than 14 without higher regularity with respect to time. However, the authors excluded the boundary condition 9 and proved the local existence solution solely in two dimensions.
In the present paper, we shall prove local existence and uniqueness solution to 6-11 such that i.a. u ∈ L 2 (0, T * ; D), D ֒→ W 2,2 (Ω) 3 , which is strong in the sense that the solutions possess second spatial derivatives. The key embedding D ֒→ W 2,2 (Ω) 3 is a consequence of assumptions setting on the domain Ω and the regularity theory for the steady Stokes system in non-smooth domains, see [19,Corollary 4.2] and [15,16].
In next Section 3 we present some auxiliary results needed in the proof of the main result stated and proved in Section 4.  where c = c(Ω).

Auxiliary results
Proof. The proof is essentially the same as the proof of Theorem 2.1 in [3].
The following result was established by Aubin (see [2]). Suppose that B 0 is continuously imbedded into B, which is also continuously imbedded into B 1 , and imbedding from B 0 into B is compact. For any given Then the imbedding from W into L p0 (0, T ; B) is compact.

4.2.
Proof of the main result.

Existence.
Remark 1. Setting w = u − u 0 this amounts to solving the problem with the homogeneous initial condition for every v ∈ V and for almost every t ∈ (0, T ), where f ∈ L 2 (0, T ; H) and u 0 ∈ D. For arbitrary fixed w ∈ X T we now consider the linear problem for every v ∈ V and for almost every t ∈ (0, T ). Definition 4.2. Let F : X T → L 2 (0, T ; H) be an operator such that for every v ∈ V and for almost every t ∈ (0, T ).
From Theorem 3.1 we deduce that for arbitrary fixed w ∈ X T there exists We now prove the following Lemma 4.3. F is a continuous operator from X T into L 2 (0, T ; H) and for all R > 0, T > 0, and for all w ∈ B R (T ) we have where C 0 (T ) → 0 + for T → 0 + and C 1 is independent of T .
Proof. Obviously, there exists C 0 (T ) > 0 such that Using the interpolation inequality one obtains Similarly, we obtain the inequalities Inequalities 41-42 and 37-38 imply that F is a continuous operator from X T into L 2 (0, T ; H).
The proof of the main result is based on the Brouwer fixed point theorem. Let the operator A be defined as follows. Given a function w ∈ X T , consider the linear problem for every v ∈ V and for almost every t ∈ (0, T ), where F is defined by Definition 4.2. Theorem 3.1 and Lemma 4.3 ensure that the linear problem 43-44 has a unique solution w ∈ Y T . Define A : X T → Y T by setting A( w) = w. Clearly, the inequality 17 and Lemma 4.3 imply that A is a continuous operator from X T into Y T . For all w ∈ B R (T ), taking 17 and 35 together, we deduce where c 3 (T ) → 0 for T → 0 and c 1 , c 2 and c 4 do not depend on T . Hence for T = T * , T * > 0 sufficiently small, and for a sufficiently large R, A maps B R (T * ) into itself. Since A is a continuous operator from X T into Y T and Y T ֒→֒→ X T , A is totally continuous operator from X T * into X T * , where X T * is a reflexive Banach space. Therefore there exists a fixed point w ∈ B R (T * ) such that A(w) = w in X T * .