On Nonhomogeneous Boundary Value Problem for the Stationary Navier-Stokes Equations in a Symmetric Cusp Domain

The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.


Introduction
Mathematically a point source is a singularity from which flux or flow is emanating. Although such singularities do not exist in the observable universe, mathematical point sources/sinks are often used as approximations to reality in physics and other fields. Point sources/sinks are often used as simple models for driving flow through a gap in a wall. In oceanography, point sources are used to model the influx of fluid from channels and holes. Another example: a nuclear explosion can be treated as a thermal point source in large-scale atmospheric simulations. This paper is devoted to the famous problem for stationary Navier-Stokes equations in the domain with multiply connected boundary. In fact, this problem arose in the pioneering Leray work (1933). Up to now, this problem is not solved for the general three dimensional case. But it has the positive solution for two dimensional (plane and axially symmetric) flows (see [7] and references there).
We assume that the support of the boundary value a ∈ W 1/2,2 (∂Ω) is separated from the cusp point O = (0, 0), i.e., be the fluxes of the boundary value a over the outer and the inner boundaries, respectively. Here n is a unit vector of the outward normal to ∂Ω. Since the fluid is incompressible, the total flux has to be zero (the necessary compatibility condition) and we have: is a cross section of G by the straight line parallel to the x 1 -axis.
Since, the velocity u has a nonzero flux in the cusp point, it has to be singular: u(x) ∼ 1/meas(σ(r)) = c/ϕ(r) → +∞ as r → 0. Moreover, the velocity u has infinite Dirichlet integral Ω |∇u| 2 dx = +∞ (infinite dissipation of energy). Therefore, it is necessary to look for the solution in a class of functions with infinite Dirichlet integral. Notice that the above formulated problem has similarities with boundary value problems for the Navier-Stokes equations in domains with paraboloidal outlets to infinity (the paraboloidal outlet to infinity in 2D has the form {x ∈ R 2 : |x 1 | ≤ ϕ(x 2 ), x 2 ∈ (H, +∞)}, where lim t→+∞ ϕ(t) = +∞). Thus, the structure of such outlet is similar to the structure of the cusp point with the difference that "singularity" is at infinity.
In multiply connected domains with outlets to infinity the solvability of nonhomogeneous boundary value problem for the stationary Navier-Stokes equations is proved either assuming the smallness condition of the fluxes over the inner boundaries (see, for instance, [3,5,12,13]) or under the certain symmetry assumptions on the domain and the boundary value (see, for instance, [1,6,10,11]). In [9,14] O.A. Ladyzhenskaya and V.A. Solonnikov proposed a method (so called Saint-Venant's estimates method) which allowed to prove the existence of solutions having infinite Dirichlet integral in domains with outlets to infinity. In the present paper we use these ideas in the case of a source or sink in the cusp point and prove the existence of a solution to problem (1.1) for arbitrarily large fluxes F (inn) and F (out) . The most important part in this prove is to construct the vector field which satisfies so called Leray-Hopf's inequalities (see (3.4)).

Main notation and auxiliary results
We will use the letter "c" for a generic constant which numerical value or dependence on parameters is unessential to our considerations; "c" may have different values in a single computation. Vector valued functions are denoted by bold letters while function spaces for scalar and vector valued functions are denoted in the same way.
Let D be a domain in R n . C ∞ (D) denotes the set of all infinitely differentiable functions defined on Ω and C ∞ 0 (D) is the subset of all functions from C ∞ (D) with compact supports in D. For given non-negative integers k and q > 1, L q (D) and W k,q (D) denote the usual Lebesgue and Sobolev spaces; W k−1/q,q (∂D) is the trace space on ∂Ω of functions from W k,q (D). J ∞ 0 (D) is the set of all solenoidal (div u = 0) vector fields u from C ∞ 0 (D). We say that Ω is a symmetric domain with respect to the x 2 -axis if the following condition is valid: (2.1) The vector function u = u 1 , u 2 defined in Ω is called symmetric with respect to the x 2 -axis if u 1 is an odd function in x 1 and u 2 is an even function in x 1 , i.e., For any set V (Ω) consisting of functions defined in the symmetric domain Ω (satisfying (2.1)), we denote by V S (Ω) the subspace of symmetric functions (satisfying (2.2)) from V (Ω).
Let Ω be a domain with a cusp point defined in Introduction. Let us introduce a family of domains Ω k with Lipschitz boundaries: We write u ∈ W l,q loc (Ω) if u ∈ W l,q (Ω k ) for ∀k. Let M be a closed set in R 2 . By ∆ M (x) we denote Stein's regularized distance from the point x to the set M. Notice that ∆ M (x) is an infinitely differentiable function in R 2 \ M and the following inequalities The positive constants a 1 , a 2 and a 3 are independent of M (see [16], Chapter VI, Sections 1 and 2, 167-171, Theorem 2).
We shall use the well known results which are formulated in the lemmas below.

Lemma 2.
Let D ⊂ R 2 be a bounded domain with Lipschitz boundary ∂D, L ⊆ ∂D, meas(L) > 0. Assume that the vector field h ∈ W 1/2,2 (∂D) satisfies the conditions L h · n dS = 0, supp h ⊆ L. Then h can be extended inside D in the form The constant c is independent of ε > 0 (see [8], Chapter V, Section 4, 127-128).
3 Solvability of problem (1.1) Definition 1. A symmetric weak solution of problem (1.1) is a solenoidal vector field u ∈ W 1,2 loc, S (Ω) satisfying the nonhomogenous boundary condition u ∂Ω\{O} = a and the integral identity Let us reduce the nonhomogeneous boundary conditions to homogeneous ones.
To do this, we need to construct a suitable extension A of the boundary value a. Since we are looking for the symmetric solution, A has to be symmetric. Moreover, it has to be solenoidal and to satisfy the condition A| ∂Ω = a. When A is constructed we can look for the solution u of problem (1.1) in the form where v ∈ W 1,2 loc, S (Ω) is a new unknown solenoidal velocity field which satisfies the homogeneous boundary condition v = 0 on ∂Ω \ {O}. Putting (3.2) into (3.1) we get the integral identity for v : The existence of v satisfying (3.3) could be proved using the ideas proposed by O.A. Ladyzenskaya and V.A Solonnikov ( [9,14]). Actually, in order to get the desirable a priori estimates, the important step is to construct the extension A satisfying so called Leray-Hopf's inequalities: for any solenoidal w ∈ W 1,2 loc, S (Ω), w = 0 on ∂Ω \ {O}. As soon as we have a suitable extension A of the boundary value a, the method of Saint-Venant's estimates can be applied and the existence of the solution can be proved. The detailed existence proof for a simply connected cusp domain can be found in [4]. This proof remains valid for the problem considering in this paper. Therefore, we just formulate the existence theorem.
Theorem 1. Suppose that Ω ⊂ R 2 is a multiply connected symmetric with respect to the x 2 -axis cusp domain and each Γ i , i = 1, ..., N, and Γ intersect the x 2 -axis (see Figure 1). Assume that the boundary value a is a symmetric vector field in W 1/2,2 (∂Ω) such that the support of a is separated from the cusp point O. Then problem (1.1) admits at least one weak solution u = A+v which satisfies the following estimate where the constant c is independent of k.

Construction of the extension
The extension A of the boundary value a will be constructed as the sum where B (inn) extends the boundary value a from the inner boundaries ∪ N i=1 Γ i and B (out) extends the modified boundary value from the outer boundary Γ. Indeed, in order to construct B (inn) , we "remove" the fluxes F Then by "removing" it to the cusp point and extending the modified boundary value from Γ into Ω we construct the extension B (out) . The vector fields B (inn) and B (out) are constructed to satisfy Leray-Hopf's inequalities which allow to obtain a priori estimates of the solutions for arbitrarily large fluxes F (inn) i , i = 1, ..., N, and F (out) . Notice that the symmetry assumption is crucial for the construction of B (inn) , satisfying Leray-Hopf's inequalities. In general case Leray-Hopf's inequalities cannot be true for the vector field B (inn) . Indeed, if the fluxes over connected components of the boundary do not vanish, there is a counterexample (see [17]) showing that in general bounded domains Leray-Hopf's inequalities can be false whatever the choice of the solenoidal extension is taken. However, such extension is possible for symmetric bounded domains (see [2]).

Construction of the extension B (inn)
In order to construct B (inn) satisfying the Leray-Hopf inequalities, we follow the idea of Fujita [2] for bounded symmetric domain.
We start with the construction of some auxiliary functions. Let 0 < κ < 1/2 be a parameter. Then we introduce non-negative functions β κ (t) : R → R with the following properties: Due to the properties of β κ (t), we see that Let δ be a small positive number. Define a smooth non-negative function Obviously, Therefore, we have that lim κ→+0 sup t |t|s(t, δ, κ) = 0.
Let us choose a small number δ so that the straight line x 1 = δ cuts each of Γ i , i = 1, ..., N at only two points. For each Γ i , i = 1, ..., N, the x 2 -axis intersects Γ i at the point (0, X i ) and (0, .., N we define the thin strips: Let us take a part of Υ i ∩ Ω which we denote by Υ i , i.e., the boundary of Υ i is the union of Γ i ∩ Υ i , [−δ, δ] × (X i + µ) and the lines x 1 = δ, x 1 = −δ. Here µ is a small positive number. Then since the vector field b (inn) i is solenoidal, we obtain Notice that the vector n denotes the unit outward normal to ∂Ω on Γ i , while the vector e 1 denotes the unit normal to ∂ Υ i on [−δ, δ] × (X i + µ), vectors n and e 1 have opposite directions. Therefore,

Moreover,
Notice that for j > i vector field b (inn) i vanishes on Γ j (by construction) and for j < i the flux of b (inn) i across Γ j cancel each other. Let us set The vector field b (inn) is symmetric and solenoidal. Moreover, for i = 1, ..., N we have Here we have used that b (inn) i vanishes on Γ j for i = j. Since condition (4.1) is valid, according to Lemma 2, there exists a solenoidal extension b satisfies the Leray-Hopf inequality: is not necessary symmetric. However, since the bound- can be sym- (4.2) Finally we define is a symmetric extension of the boundary value a from ∪ N i=1 Γ i . It remains to prove that B (inn) satisfies the Leray-Hopf inequalities.

Construction of the extension B (out)
After the construction of the extension B (inn) of the boundary value a from the inner boundaries Γ 1 , ..., Γ N , we need to construct an extension B (out) which extends the modified boundary value a − b (inn) from Γ .
is solenoidal, it is L 2 -orthogonal to the first term of the right hand side of (4.4).

Let us define Ω
We start with the construction of the vector field b (out) + in the domain Ω + . Take any point x + ∈ Λ + and introduce a smooth simple curve γ + = l + ∪ γ + 0 , where l + = {x 1 = 0 : 0 ≤ x 2 ≤ H} and γ + 0 ⊂ Ω + 0 connects the line l + with the point x + . The curve γ + does not intersect any inner boundary Γ 1 , ..., Γ N (see Fig.3). We define a cut-off function ξ + by the formula: where Ψ is a smooth function: . Moreover, the following inequalities , hold with the constant c independent of ε.
Proof. The first statement follows directly from the definition of the function Ψ (t). Estimates (4.5) follow by direct computations using the properties of the regularized distance and the fact that supp ∇ξ + is contained in the set where ∆ ∂Ω + \Λ + (x) ≤ ∆ γ + (x) (see for details the proof of Lemma 2 in [15]).
Since the curve γ + divides Ω + into two parts, we define ξ + (x, ε) = ξ + (x, ε) for points laying on the right hand side of the curve γ + and ξ + (x, ε) = 0 for points laying on the left hand side of the curve γ + . Then we introduce the vector field b hold with the constant c in (4.7) independent of ε.
Proof. The first statement follows directly from the definition of the vector field b (out) + and Lemma 4. Since div b (out) + = 0 and due to properties of ξ + , we have Using estimates (4.5) and definition (4.6), we derive and . (4.11) Since for points , we obtain (using the properties of the regularized distance) (4.12) where c 1 and c 2 are positive constants. Finally, the estimates (4.8) and (4.9) follow from (4.10), (4.11) and (4.12) (see [4] for details).
Proof. Applying the Hölder inequality and estimates (4.7), (2.3) we obtain The same argument proves the Leray-Hopf inequality in ω + k .
Notice that the vector n denotes the unit outward normal to ∂Ω on Γ , while the vector e 1 denotes the unit normal to ∂ Υ on [−δ, δ] × (X 0 − µ), vectors n and e 1 have the opposite directions. Therefore, (4.14) From (4.13) and (4.14) we have that Because of Λ h · n dS = 0, according to Lemma 2, there exists an extension b .
Analogously we get the estimate (4.21).
According to Lemma 3 and Lemma 7, the constructed vector field has all the necessary properties that insure the validity of Theorem 1 formulated in Section 3.