Non-stationary Navier–Stokes equations in 2D power cusp domain: II. Existence of the solution

Konstantin Pileckas; Alicija Raciene

doi:10.1515/anona-2020-0165

Open Access Published by De Gruyter February 28, 2021

Non-stationary Navier–Stokes equations in 2D power cusp domain

II. Existence of the solution

Konstantin Pileckas and Alicija Raciene

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2020-0165

Abstract

The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. We consider the case where the boundary value has a nonzero flux over the boundary. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point was constructed. In this, second part, the constructed asymptotic decomposition is justified, i.e., existence of the solution which is represented as the sum of the constructed asymptotic expansion and a term with finite energy norm is proved. Moreover, it is proved that the solution represented in this form is unique.

Keywords: Nonstationary Navier-Stokes problem; power cusp domain; singular solutions; asymptotic expansion

MSC 2010: 35Q30; 35A20; 76M45; 76D03; 76D10

1 Introduction

In this paper we continue to study the boundary value problem for the non-stationary Navier–Stokes system

(1.1) ut−νΔu+(u⋅∇)u+∇p=f,divu=0,u|∂Ω∖O=a,u(x,0)=b(x)

in a two-dimensional bounded domain Ω with the cusp point O = (0, 0) at the boundary: Ω = G_H ∪ Ω₀, where GH=x∈R2:|x1|<φ(x2),x2∈(0,H],φ(x2)=γ0x2λ,γ0=const,λ>1 (see Figure 1). For simplicity we assume that the boundary ∂Ω ∩ ∂Ω₀ is infinitely smooth. Here u = (u₁, u₂) stands for the velocity field, p stands for the pressure, ν > 0 is the constant kinematic viscosity.

Fig. 1

Domain Ω

It is supposed that supp a ⊂ ∂Ω₀ ∩ ∂Ω, i.e., the support of the boundary value a∈L2(0,T;W3/2,2(∂Ω)) is separated from the cusp point O. We also assume that the flux F (t) of a is nonzero:

(1.2) ∫∂Ωa⋅ndS=−F(t),F(t)/≡0,F(0)=0.

The initial velocity b ∈ W^1,2(Ω) and the boundary value a have to satisfy the necessary compatibility conditions

(1.3) divb(x)=0,b(x)|∂Ω=a(x,0).

The solution u of (1.1) has to satisfy the condition

∫σ(h)u⋅ndS+∫∂Ω∩∂Ω0a⋅ndS=0

where σ(h) = {x ∈ G_H : x₂ = h = const}, which means that the total flux of the fluid is equal to zero. Thus,

(1.4) ∫σ(h)u⋅ndx1=F(t)/≡0,

and we can regard the cusp point O as a source (or a sink) of intensity F (t).

More information and references concerning the Navier–Stokes equations in domains with singular boundaries are given in the introduction to the first part of the paper, see [1]. In [1] the formal asymptotic decomposition of the solution (u, p) near the cusp point was constructed. This asymptotic expansion has the form

(1.5) U[J]x1x2λ,x2,t,τ=UO,[J]x1x2λ,x2,t+UB,[J]x1x2λ,x2,τ,P[J]x1x2λ,x2,t,τ=PO,[J]x1x2λ,x2,t+PB,[J]x1x2λ,x2,τ,

where the pair (U^O^,[J], P^O^,[J]) is an approximate solution (outer asymptotic expansion) of the Navier–Stokes problem in variables y1=x1x2−λ,y2=x2,t=t; the "slow" time variable t plays the role of a parameter and, in general, the initial condition is not satisfied. The pair (U^B^,[J], P^B^,[J]) is the boundary layer corrector (the inner part of the asymptotic expansion) which compensate the discrepancy in the initial condition. Notice that (U^B^,[J], P^B^,[J]) exponentially vanishes as the fast time τ=tx22λ tends to infinity. The number J is taken so large that the discrepancy H[J]of(U[J],P[J]) in the Navier–Stokes equations belongs to the space L²(0, T; L²(Ω)), while the discrepancy in the initial condition is zero (see [1] for details). Moreover, in order to ensure the existence of all terms of asymptotic decomposition up to the order J, we have to assume that

∂la∂tl∈L2(0,T;W1/2,2(∂Ω)),l=1,2,...,J+1.

Here we justify this asymptotics. We prove that there exists a solution of problem (1.1) which is represented as a sum of the singular part (the constructed asymptotic decomposition) and the function having finite energy. To be more precise, we construct a solenoidal extension V of the boundary value a which coincides with U^[J] near the cusp point and we look for the solution (u, p) of (1.1) in the form u = V + v, p = ζP^[J] + q, where ζ is a smooth cut-off function localising the asymptotical part of the pressure near the cusp point O. Then for (v, q) we obtain the problem

(1.6) vt−νΔv+(v⋅∇)v+(V⋅∇)v+(v⋅∇)V+∇q:=fˆ,divv=0,v|∂Ω=0,u(x,0)=b(x)−V(x,0):=uˆ0(x)

with fˆ∈L2(0,T;L2(Ω)),uˆ0∈W∘ 1,2(Ω). In the paper we prove the existence of a unique regular solution v to (1.6).

The existence of singular solutions to the time-periodic and the non-stationary Stokes problem in the domain with a cusp point was studied in [2, 3]. We can also mention the recent paper [4] where the Dirichlet problem for the non-stationary Stokes system is studied in a three-dimensional cone. The non-stationary Navier–Stokes equations in tube-structures were studied in [5, 6]. The solvability of the stationary Navier–Stokes system in the cusp domain with source or sink in the cusp point was proved in [7]. The steady Navier-Stokes equations are also studied in a punctured domain Ω = Ω₀ \ {O} with O ∈ Ω₀ assuming that the point O is a sink or source of the fluid, see [8, 9, 10] and [11] for the review of these results. We also mention the papers [12, 13, 14, 15] where the stationary Navier–Stokes equations were studied in domains with paraboloidal outlets to infinity. Such geometry has similarities with the cusp domains, the difference is that in the case of a domain with outlet to infinity x₂ → ∞, while in the cusp domain x₂ → 0.

The paper is organised as follows. In Section 2 we introduce the main notation, function spaces and prove certain inequalities needed in subsequent sections. In Section 3 we study the Stokes problem and the Stokes operator in the cusp domain. Finally, the main result of the paper, the unique solvability of problem (1.6), is proved in Section 4.

2 Notation, function spaces and auxiliary results

Let G be a domain in Rⁿ. We use usual notation of functional spaces (e.g., [16]). By L^p(G) and W^m^,p(G), 1 ≤ p < ∞, we denote the Lebesgue and Sobolev spaces, respectively. The norm in a Banach space X is denoted by ∥⋅∥X.C∞(G) is the set of all infinitely differentiable functions defined on G and C0∞(G) - the subset of all functions from C∞(G) having compact supports in G. By W∘ k,q(G) we denote the completion of the C0∞(G) in the ∥⋅∥Wm,p(G) -norm and by W^m^−1/p,p(∂G) the space of traces on ∂G of functions from W^m^,p(G). The space L^p(0, T; X) consists of all measurable functions u : [0, T] → X with

∥u∥Lp(0,T;X)=∫0T∥u(t)∥Xpdt1/p<∞,1≤p<∞.

We do not distinguish in notation the spaces of vector and scalar functions; from the context it will be clear which space we have in mind.

Denote J0∞(G)={v∈C0∞(G):divv=0} the set of all divergence free vector fields from C0∞(G) and by J₀(G) be the closure of J0∞ (G) in L²(G)-norm. Let H(G)={v∈W∘1,2(G): div v = 0}. If G is a bounded domain with Lipschitz boundary, then H (G) coincides with the closure of J0∞(G) in the norm ∥⋅∥W1,2(G) (see [17]).

Let us consider the cusp domain Ω. Let h₀ = H, hk=hk−1−φ(hk−1)2L, where L is a Lipschitz constant for the function φ, k = 1,2, . . . . The sequence {h_k} is decreasing and bounded from below. Assume that the limit of this sequence is a₀ ≠ 0. From the definition of the sequence it follows that a0=a0−φ(a0)2L. Then φ(a₀) = 0. However, φ(a₀) ≠ 0 for a₀ ≠ 0 and, hence the limit a₀ = 0. Since the sequence is decreasing and the limit is equal to 0, all its elements are positive.

Denote ωl={x∈R2:|x1|<φ(x2),x2∈(hl,hl−1)},l=1,…. Note that

(2.1) 12φ(hl)≤φ(t)≤32φ(hl),t∈[hl+1,hl].

Define the transformation y = P_lx by the formulas

(2.2) y1=2Lx1φ(x2),y2=2L(x2−hl−1)φ(hl−1)

and introduce the domains

G0={y∈R2:|y1|<2L,−1<y2<0},G1={y∈R2:|y1|<2L,−1−g(y1)<y2<−1},G2={y∈R2:|y1|<2L,−2<y2<1}.

In the definition of G₁ the function g ∈ C^∞ satisfies the conditions g(±2L) = 0,0 < g(y₁) < 1 for |y₁| < 2L and it is such that the curve {y:y1=−2L}∪{y:y1=2L}∪{y:|y1|<2L,y2=−1−g(y1)} is infinitely smooth.

Obviously the transformation Pl−1 maps G₀ onto ω_l. Consider the domain ωl∗=Pl−1G2. Then ωl⊂ωl∗= {x∈R2:|x1|<φ(x2),hl−1−φ(hl−1)L<x2<hl−1+φ(hl−1)2L}⊂ωl+1∪ωl∪ωl−1,l=2,3,… (see Figure 2). It is easy to see that

Ω=Ω0∪(⋃l=1∞ωl)=Ω0∪ω1∪(⋃l=2∞ωl∗).

$Fig. 2 Domains ωj,ωj∗andωˆj+1. $ \omega_j, \omega_j^* \;{\text {and}}\; \hat\omega_{j+1}.$ in different coordinate systems$

Fig. 2

Domains ωj,ωj∗andωˆj+1. in different coordinate systems

Let us fix K ≥ 2 (sufficiently large) and define ωˆK=PK−1−1G1,ωˆK∗=PK−1−1(G0∪G1). Obviously, ωˆK⊂ωK and ωˆK⊂ωˆK∗. Let

ΩK=Ω0∪(⋃l=1K−1ωl)∪ωˆK.

The boundary of Ω_K consist of ∂Ω∩ΩˉK and the curve Γ_K which is defined as ΓK=PK−1−1E0, where E₀ = {y : |y₁| < 2L, y₂ = −1 − g(y₁)}. By construction, the boundary ∂Ω_K is smooth (see Figure 3).

We can take also the other covering of the domain Ω_K. Namely,

ΩK=Ω0∪ω1∪(⋃l=1K−2ωl)∪ωˆK−1∗∪ωˆK∗,

where ωˆK−1∗=PK−1−1({y∈R2:|y1|<2L,−1−g(y1)<y2<−1})=PK−1−1(G1).

We also introduce domains Ωl♯=Ω0⋃(⋃j=1lωj),l=1,2,… (see Figure 4).

In Ω_K we define the function space L12(ΩK) with the weighted norm

∥f∥L12(ΩK)2=∫Ω0|f(x)|2dx+∫ΩK∖Ω0φ2(x2)|f(x)|2dx.

Let us prove some auxiliary inequalities for functions defined in Ω.

Fig. 3

Domains Ω_K.

$Fig. 4 Domains Ωj♯ $ {\it\Omega}_{j}^{\sharp}$$

Fig. 4

Domains Ωj♯

Lemma 2.1

(Poincaré inequalities). Let u∈Wloc1,2(Ωˉ),u|∂Ω=0. Then the following inequalities

(2.3) ∫ωl|u(x)|2dx≤9π2φ2(hl−1)∫ωl|∇u(x)|2dx,

(2.4) ∫hH∫−φ(x2)φ(x2)|φ(x2)|κ−2|u(x)|2dx≤4π2∫hH∫−φ(x2)φ(x2)|φ(x2)|κ|∇u(x)|2dx

hold for any κ ∈ R and any h ∈ (0, H).

Proof

By the classical Poincaré inequality on the interval ( − φ(x₂), φ(x₂)), we have

∫−φ(x2)φ(x2)|u(x1,x2)|2dx1≤4π2φ2(x2)∫−φ(x2)φ(x2)|∂u(x1,x2)∂x1|2dx1.

Integrating this inequality over (h_l , h_l₋₁) and applying (2.1) we derive (2.3). To prove (2.4) it is enough to multiply the above inequality by |φ(x₂)|^κ⁻² and integrate over the interval (h, H).

Lemma 2.2

Let u∈Wloc1,2(Ωˉ). Then

(2.5) ∥u∥L4(ωl)4≤cφ−2(hl−1)∥u∥L2(ωl)2(∥u∥L2(ωl)2+φ2(hl−1)∥∇u∥L2(ωl)2)

holds with a constant c independent of l. In particular, if u∈Wloc2,2(Ωˉ), then the following estimate

holds.

∥∇u∥L4(ωl)4≤cφ−2(hl−1)∥∇u∥L2(ωl)2(∥∇u∥L2(ωl)2+φ2(hl−1)∥∇2u∥L2(ωl)2)

Proof

After the transformation P_l, the domain ω_l is transformed into the domain G0={y:|y1|<1,−1< y2<0} which is independent of l. In G₀ holds the inequality (see [18])

∥u∥L4(G0)4≤c∥u∥L2(G0)2(∥u∥L2(G0)2+∥∇u∥L2(G0)2).

Passing in the last inequality to variables x we obtain

∫ωl|u|44L2φ(x2)φ(hl−1)dx≤c(∫ωl|u|24L2φ(x2)φ(hl−1)dx)(∫ωl|u|24L2φ(x2)φ(hl−1)dx+∫ωl|∇yu|24L2φ(x2)φ(hl−1)dx)

≤c∫ωl|u|24L2φ(x2)φ(hl−1)dx(∫ωl|u|24L2φ(x2)φ(hl−1)dx+∫ωl[|∂u∂x1|2(φ2(x2)+|x1|2φ2(hl−1)φ2(x2)|φ′(x2)|2)+|∂u∂x2|2φ2(hl−1)]4L2φ(x2)φ(hl−1)dx).

Since |x₁| < φ(x₂) and |φ′(x₂)| ≤ const, from the last inequality using (2.1) we derive (2.5). From (2.3) and (2.5) we obtain

Lemma 2.3

Let u∈Wloc1,2(Ωˉ),u|∂Ω=0. Then

(2.6) ∥u∥L4(ωl)≤c∥u∥L2(ωl)1/2∥∇u∥L2(ωl)1/2≤cφ1/2(hl−1)∥∇u∥L2(ωl)

with a constant c is independent of l.

Let us consider in ω_l the divergence problem

(2.7) divv=gin ωl,v=0on ∂ωl.

Lemma 2.4

Let g ∈ L²(ω_l) and

(2.8) ∫ωlg(x)dx=0.

Then there exists a solution v ∈ W˚^1,2(ω_l) of (2.7) satisfying the estimate

(2.9) ∥v∥L2(ωl)≤c∥g∥L2(ωl)

with a constant c independent of l.

Proof

The transformation P_l (see (2.2)) maps the domain ω_l onto G₀ = {y : |y₁| < 1,−1 < y₂ < 0}. Because of (2.8),

∫G0g(Pl−1(y))Jl−1(Pl−1(y))dy=∫ωlg(x)dx=0,

where Jl(x)=4L2φ(hl−1)φ(x2) is the Jacobian. Therefore (see [17]), there exists a function vˆ∈W∘ 1,2(G0) such that

divyvˆ(y)=Jl−1(Pl−1(y))g(Pl−1(y))

and

(2.10) ∥∇yvˆ∥L2(G0)≤c∥Jˆl−1gˆ∥L2(G0),

where gˆ(y)=g(Pl−1(y)), etc. Let us define the vector field v(x) with the components

v1(x)=vˆ1(y)|y=Pl(x)2Lφ(hl−1)+vˆ2(y)|y=Pl(x)2Lx1φ′(x2)φ2(x2),v2(x)=vˆ2(y)|y=Pl(x)2Lφ(x2).

Then it is straightforward to verify that

divxv=4L2φ(hl−1)φ(x2)divyvˆ(y)|y=Pl(x)=g(x).

Estimating now ∥∇yvˆ∥L2(G0)2 by inequality (2.10), using the expression for the Jacobean and returning to coordinates x, we derive estimate (2.9) with a constant independent of l.

Remark 2.1

It is easy to see that Lemmas 2.1–2.4 remain valid if we take the domains ωl∗,l=2,…,k,orωˆk, ωˆk∗ instead of ω_l.

3 Stokes problem and Stokes operator

3.1 Estimates of solutions to the Stokes problem

In Ω_K consider the Dirichlet boundary value problem for the Stokes system

(3.1) −νΔv+∇p=f,divv=0,v|∂ΩK=0.

The weak solution v ∈ H (Ω_K) to (3.1) satisfies the integral identity

ν∫ΩK∇v⋅∇ηdx=∫ΩKf⋅ηdx∀η∈H(ΩK).

Lemma 3.1

Let f∈L12(ΩK). Then problem (3.1) has a unique solution v ∈ H (Ω_K) and there holds the estimate

(3.2) ∥∇v∥L2(ΩK)≤c∥f∥L12(ΩK)

with a constant c independent of K.

Proof

By Poincaré’s inequality (2.4) with κ = 0,

|∫ΩKf⋅ηdx|≤c∥f∥L12(ΩK)2∥∇η∥L2(ΩK).

Hence, the statement of the lemma follows from Lax–Milgram’s theorem.

Lemma 3.2

Let f∈L2(ΩK)⊂L12(ΩK). Then the weak solution v of (3.1) satisfies the estimate

(3.3) ∫Ω0|∇v|2dx+∫ΩK∖Ω0φ−2(x2)|∇v|2dx≤c∫ΩK|f|2dx

with a constant c independent of K.

Proof

Let

Φ(x2)=1φ2(hl),x∈Ω0⋃(⋃j=1lωj)=Ωl♯,1φ2(x2),x∈(⋃j=l+1K−1ωj)⋃ωˆK=ΩK∖Ωl♯.

Here and below the number l is fixed; we specify it during the proof.

Consider the function u = Φ(x₂)v. Then div u = Φ′(x₂)v₂. Since v ∈ H (Ω_K), the flux of v over any section x₂ = const of G_H is equal to zero, i.e.,

∫−φ(x2)φ(x2)Φ′(x2)v2(x1,x2)dx1=0∀x2∈(hK,H].

Integrating over x₂ we conclude that

∫ωjΦ′(x2)v2(x1,x2)dx=0,j=l+1,…,K−1;∫ωˆKΦ′(x2)v2(x1,x2)dx=0.

Then by Lemma 2.4, there exist functions wj∈W∘ 1,2(ωj),j=l+1,…,K−1,wK∈W∘ 1,2(ωˆK) such that

divwj=−Φ′(x2)v2inωj,j=l+1,…,K−1,divwK=−Φ′(x2)v2inωˆK.

Moreover, the following estimates

(3.4) ∥∇wj∥L2(ωj)≤c∥Φ′v2∥L2(ωj)≤cmaxx∈ωj|φ′(x2)|∥φ−3v2∥L2(ωj),j=l+1,…,K−1,∥∇wK∥L2(ωˆK)≤cmaxx∈ωˆK|φ′(x2)|∥φ−3v2∥L2(ωˆK)

hold with a constant c independent of j and K. Taking into account inequalities (2.1), from (3.4) we obtain

(3.5) ∥φ(x2)∇wj∥L2(ωj)≤cmaxx∈ωj|φ′(x2)|∥φ−2(x2)v2∥L2(ωj),j=l+1,…,K−1,∥φ(x2)∇wK∥L2(ωˆK)≤cmaxx∈ωˆK|φ′(x2)|∥φ−2(x2)v2∥L2(ωˆK).

Define the function

w(x)=0,x∈Ωl♯,wj(x),x∈ωj,j=l+1,…,K−1,wK(x),x∈ωˆK

(recall that Φ′(x2)=0inΩl♯). Take in the integral identity η = u + w. By construction, div (u + w) = 0 and hence, u + w ∈ H (Ω_K). This yields

ν∫ΩKΦ|∇v|2dx=−ν∫ΩK∇v⋅∇Φ⋅vdx−ν∫ΩK∇v⋅∇wdx+∫ΩKf⋅(Φv+w)dx=J1+J2+J3.

Let us estimate the integrals J_i in the right-hand side of the last relation. Using (2.4) and (3.5) we get

|J1|=|ν∫ΩK∖Ωl♯∇v⋅∇Φ⋅vdx|≤2ν∫ΩK∖Ωl♯|∇v||v||φ′(x2)|φ−3(x2)dx≤csupx∈ΩK∖Ωl♯|φ′(x2)|(∫ΩK∖Ωl♯φ−2(x2)|∇v|2dx)1/2(∫ΩK∖Ωl♯φ−4(x2)|v|2dx)1/2≤csupx∈ΩK∖Ωl♯|φ′(x2)|∫ΩK∖Ωl♯φ−2(x2)|∇v|2dx;|J2|≤ν∑j=l+1K−1|∫ωj∇v⋅∇wjdx|+|∫ωˆK∇v⋅∇wKdx|≤ε∑j=l+1K−1∫ωjφ−2(x2)|∇v|2dx+ε∫ωˆKφ−2(x2)|∇v|2dx+cε∑j=l+1K−1∫ωjφ2(x2)|∇wj|2dx+cε∫ωˆKφ2(x2)|∇wK|2dx≤ε∫ΩK∖Ωl♯φ−2(x2)|∇v|2dx+cεsupx∈ΩK∖Ωl♯|φ′(x2)|2(∑j=l+1K−1∫ωjφ−2(x2)|∇v|2dx+∫ωˆKφ−2(x2)|∇v|2dx)≤(ε+cεsupx∈ΩK∖Ωl♯|φ′(x2)|2)∫ΩK∖Ωl♯φ−2(x2)|∇v|2dx;

|∫ΩKf⋅Φvdx|≤1φ2(hl)|∫Ωl♯f⋅vdx|+∑j=l+1K−1|∫ωjf⋅1φ2(x2)vdx|+|∫ωˆKf⋅1φ2(x2)vdx|≤cε(1φ2(hl)∫Ωl♯|f|2dx+∑j=l+1K−1∫ωj|f|2dx+∫ωˆK|f|2dx)+ε(1φ2(hl)∫Ωl♯|v|2dx+∑j=l+1K−1∫ωj1φ4(x2)|v|2dx+∫ωˆK1φ4(x2)|v|2dx)≤c∫ΩK|f|2dx+ε(cφ2(hl)∫Ωl♯|∇v|2dx+∑j=l+1K−1∫ωj1φ2(x2)|∇v|2dx+∫ωˆK1φ2(x2)|∇v|2dx)≤c∫ΩK|f|2dx+cε∫ΩKΦ|∇v|2dx;|∫ΩKf⋅wdx|=|∫ΩK∖Ωl♯f⋅wdx|≤∑j=l+1K−1|∫ωjf⋅wjdx|+|∫ωˆKf⋅wKdx|≤c∫ΩK|f|2dx+∑j=l+1K−1∫ωj|wj|2dx+∫ωˆK|wK|2dx≤c∫ΩK|f|2dx+∑j=l+1K−1∫ωjφ2(x2)|∇wj|2dx+∫ωˆKφ2(x2)|∇wK|2dx≤c∫ΩK|f|2dx+c∑j=l+1K−1supx∈ωj|φ′(x2)|2∫ωjφ−4(x2)|v2|2dx+csupx∈ωˆK|φ′(x2)|2∫ωˆKφ−4(x2)|v2|2dx≤c∫ΩK|f|2dx+csupx∈ΩK∖Ωl♯|φ′(x2)|2∫ΩK∖Ωl♯φ−2(x2)|∇v|2dx.

Collecting the obtained estimates yields

(3.6) ν∫ΩKΦ|∇v|2dx≤c1∫ΩK|f|2dx+c2(ε+supx∈ΩK∖Ωl♯|φ′(x2)|)∫ΩKΦ|∇v|2dx,

where the constant c₁ is independent of K (but c₁ depends on l) and c₂ is independent of K and l. The function φ′(x₂) is monotonically decreasing and tends to zero as x₂ → 0. Hence supx∈ΩK∖Ωl♯|φ′(x2)|=|φ′(hl)| and liml→∞φ′(hl)=0. We choose and fix l such that |φ′(hl)|≤ν/(4c2) and take ε=ν/(4c2). Then from (3.6) follows the inequality

∫ΩKΦ|∇v|2dx≤c1∫ΩK|f|2dx

which is equivalent to (3.3).

Lemma 3.3

For sufficiently large K the weak solution v of problem (3.1) satisfies the estimate

(3.7) ∥∇2v∥L2(ΩK)≤c∥f∥L2(ΩK)

with a constant c independent of K.

Proof

Consider the solution *(v, p) of problem (3.1) in the domain ωl∗,l=2,…,K−2. Changing the variables y=Pl(x) (see (2.2)) we rewrite problem (3.1) in coordinate y in the domain G₂:

(3.8) −νΔyvˆ(y)+∇yqˆ(y))=φ2(hl−1)4L2fˆ(y)+Hˆ(y),divyv˜(y)=gˆ(y),v˜|y1=±2L=0,

where vˆ(y)=v(Pl−1y),etc.,qˆ(y)=φ(hl−1)2Lpˆ(y),

(3.9) Hˆ(y)=α11(y)∂2vˆ∂y12+α12(y)∂2vˆ∂y1∂y2+β1(y)∂vˆ∂y1+γ(y)∂qˆ∂y1,α11(y)=ν((φ2(hl−1)−φ2(x2))φ2(x2)+x12φ′(x2)φ2(hl−1)φ4(x2))|x=Pl−1(y),α12(y)=−ν2x1φ′(x2)φ(hl−1)φ2(x2)|x=Pl−1(y),β1(y)=−νx1(φ′′(x2)φ(x2)−2φ′2(x2))φ2(hl−1)2Lφ3(x2)|x=Pl−1(y),γ(y)=(γ1(y),γ2(y))T,γ1(y)=−φ(hl−1)−φ(x2)φ(x2)|x=Pl−1(y),γ2(y)=−x1φ′(x2)φ(hl−1)φ2(x2)|x=Pl−1(y),gˆ(y)=μ1(y)∂vˆ1∂y1+μ2(y)∂vˆ2∂y1,μ1(y)=φ(x2)−φ(hl−1)φ(x2)|x=Pl−1(y),μ2(y)=x1φ′(x2)φ(hl−1)φ2(x2)|x=Pl−1(y).

Applying the usual local ADN-estimates for elliptic problems (see [19, 20]) in the pair of domains G₀ ⊂ G₂, we obtain the estimate

(3.10) ∥vˆ∥L2(G0)2+∥∇yvˆ∥L2(G0)2+∥∇y2vˆ∥L2(G0)2+∥∇yqˆ∥L2(G0)2≤c(φ4(hl−1)∥fˆ∥L2(G2)2+∥Hˆ∥L2(G2)2+∥gˆ∥W1,2(G2)2+∥vˆ∥L2(G2)2+∥qˆ−qˉ∥L2(G2)2),

where qˉ=1|G2|∫G2qˆ(y)dy.Since∫G2(qˆ(y)−qˉ)dy=0, there exists w∈W∘ 1,2(G2) such that div yw=qˆ(y)−qˉ in G₂ and

∥∇yw∥L2(G2)≤c∥qˆ−qˉ∥L2(G2)

(see [17]). Multiplying (3.8) by w and integrating by parts yields

∥qˆ−qˉ∥L2(G2)2=∫G2qˆ(y)(qˆ(y)−qˉ)dy=∫G2qˆ(y)divywdy=ν∫G2∇yvˆ⋅∇ywdy−φ2(hl−1)4L2∫G2fˆ⋅wdy−∫G2Hˆ⋅wdy≤ν∥∇yvˆ∥L2(G2)∥∇yw∥L2(G2)+φ2(hl−1)4L2∥fˆ∥L2(G2)∥w∥L2(G2)+∥Hˆ∥L2(G2)∥w∥L2(G2)≤c∥∇yvˆ∥L2(G2)∥qˆ−qˉ∥L2(G2)+cφ2(hl−1)∥fˆ∥L2G2)∥qˆ−qˉ∥L2(G2)+c∥Hˆ∥L2(G2)∥qˆ−qˉ∥L2(G2).

Therefore,

(3.11) ∥qˆ−qˉ∥L2(G2)≤c(∥∇yvˆ∥L2(G2)+φ2(hl−1)∥fˆ∥L2(G2)+∥Hˆ∥L2(G2)).

From (3.10) using (3.11) and Poincaré’s inequality (2.3) we derive

(3.12) ∥vˆ∥L2(G0)2+∥∇yvˆ∥L2(G0)2+∥∇y2vˆ∥L2(G0)2+∥∇yqˆ∥L2(G0)2≤c(φ4(hl−1)∥fˆ∥L2(G2)2+∥Hˆ∥L2(G2)2+∥gˆ∥W1,2(G2)2+∥∇yvˆ∥L2(G2)2).

By definition ωl∗={x:|x1|≤φ(x2),hl−1−φ(hl−1)L<x2<hl−1+φ(hl−1)2L} and the following inequality

|φ(hl−1)−φ(x2)|≤maxx2∈ωl∗|φ′(x2)||hl−1−x2|≤cmaxx2∈ωl∗|φ′(x2)|φ(hl−1)

holds. For y=Pl(x),x∈ωl∗, using this inequality, (2.1) and the definition φ(x2)=γ0x2λ,λ>1, we obtain

|α11(y)|+|α12(y)|+|β1(y)|+|γ1(y)|+|γ1(y)|+|μ1(y)|+|μ2(y)|≤ε(l),

where liml→∞ε(l)=0 see (3.9)). Therefore, from (3.12) follows the estimate

∥vˆ∥L2(G0)2+∥∇yvˆ∥L2(G0)2+∥∇y2vˆ∥L2(G0)2+∥∇yqˆ∥L2(G0)2≤c(φ4(hl−1)∥fˆ∥L2(G2)2+∥∇yvˆ∥L2(G2)2)+cε(l)(∥∇y2vˆ∥L2(G2)2+∥∇yqˆ∥L2(G2)2).

Passing to coordinates x and using the same arguments we derive

(3.13) 1φ2(hl−1)∥v∥L2(ωl)2+∥∇xv∥L2(ωl)2+φ2(hl−1)∥∇x2v∥L2(ωl)2+∥∇xq∥L2(ωl)2≤c(φ2(hl−1)∥f∥L2(ωl∗)2+∥∇xv∥L2(ωl∗)2)+cε(l)(φ2(hl−1)∥∇x2v∥L2(ωl∗)2+∥∇xq∥L2(ωl∗)2),l=1,…,K−2.

The constant c in (3.13) is independent of l. Multiplying (3.13) by 1φ2(hl−1) yields

(3.14) 1φ4(hl−1)∥v∥L2(ωl)2+1φ2(hl−1)∥∇xv∥L2(ωl)2+∥∇x2v∥L2(ωl)2+∥∇xp∥L2(ωl)2≤c(∥f∥L2(ωl∗)2+1φ2(hl−1)∥∇xv∥L2(ωl∗)2)+cε(l)(∥∇x2v∥L2(ωl∗)2+∥∇xp∥L2(ωl∗)2),l=1,…,K−2.

By the same ADN-estimate together with the properties of the domain ωˆK , we get the inequalities

(3.15) 1φ4(hK−2)∥v∥L2(ωK−1)2+1φ2(hK−2)∥∇xv∥L2(ωK−1)2+∥∇x2v∥L2(ωK−1)2+∥∇xp∥L2(ωK−1)2≤c(∥f∥L2(ωK−1∗)2+1φ2(hK−2)∥∇xv∥L2(ωK−1∗)2)+cε(K)(∥∇x2v∥L2(ωK−1∗)2+∥∇xp∥L2(ωK−1∗)2)

and

(3.16) 1φ4(hK−1)∥v∥L2(ωK)2+1φ2(hK−1)∥∇xv∥L2(ωk)2+∥∇x2v∥L2(ωK)2+∥∇xp∥L2(ωK)2≤c(∥f∥L2(ωK∗)2+1φ2(hK−1)∥∇xv∥L2(ωK∗)2)+cε(K)(∥∇x2v∥L2(ωK∗)2+∥∇xp∥L2(ωK∗)2),

with constants independent of K.

Let l₀ < K−2 be a positive natural number (l₀ be fixed later). Arguing as above we can prove the following local estimate for the pair of domains Ωl0+1♯⊂Ωl0+2♯:

(3.17) ∥v∥W2,2(Ωl0+1♯)2+∥∇p∥L2(Ωl0+1♯)2≤c(∥f∥L2(Ωl0+2♯)2+∥∇xv∥L2(Ωl0+2♯)2).

Summing inequalities (3.14) from l₀ to K −2, adding (3.15)-(3.17) and taking into account that φ(hl−1)∼φ(x2) in ω_l and ωl∗ (see (2.1)), we get

(3.18) ∥v∥W2,2(Ωl0+1♯)2+∥∇p∥L2(Ωl0+1♯)2+∥φ−2v∥L2(ΩK∖Ωl0+1♯)2+∥φ−1∇v∥L2(ΩK∖Ωl0+1♯)2+∥∇2v∥L2(ΩK∖Ωl0+1♯)2+∥∇p∥L2(ΩK∖Ωl0+1♯)2≤c1(∥f∥L2(ΩK)2+∥∇v∥L2(Ωl0+2♯)2+∥φ−1∇v∥L2(ΩK∖Ωl0♯)2)+c2ε(l0)(∥∇2v∥L2(ΩK∖Ωl0♯)2+∥∇p∥L2(ΩK∖Ωl0♯)2).

Since liml→∞ε(l)=0, we can chose l₀ to satisfy c₂ε(l₀)≤ κ, where κ is such that

κ(∥∇2v∥L2(ΩK∖Ωl0♯)2+∥∇p∥L2(ΩK∖Ωl0♯)2)≤12(∥∇2v∥L2(Ωl0+1♯)2+∥∇p∥L2(ΩK)2+∥∇2v∥L2(ΩK∖Ωl0♯)2).

Then estimate (3.18) takes the form

∥v∥W2,2(Ωl0+1♯)2+∥∇p∥L2(Ωl0+1♯)2+∥φ−2v∥L2(ΩK∖Ωl0+1♯)2+∥φ−1∇v∥L2(ΩK∖Ωl0+1♯)2+∥∇2v∥L2(ΩK∖Ωl0+1♯)2+∥∇p∥L2(ΩK∖Ωl0+1♯)2≤c1(∥f∥L2(ΩK)2+∥∇v∥L2(Ωl0+2♯)2+∥φ−1∇v∥L2(ΩK∖Ωl0♯)2).

In particular,

(3.19) ∥∇2v∥L2(ΩK)2≤c1(∥f∥L2(ΩK)2+∥∇v∥L2(Ωl0+2♯)2+∥φ−1∇v∥L2(ΩK∖Ωl0♯)2).

Estimating the last two term in the right-hand side of (3.19) by (3.2) and (3.3) we obtain (3.7).

3.2 Stokes operator

The most results we present in this subsection are standard (e.g., [21]). Problem (3.1) can be rewritten in the operator form (without loss of generality we suppose that f ∈ J₀(Ω_K)^[1], adding the gradient part to the pressure)

Δ˜v=f,

where Δ˜=PΔ:H(ΩK)∩W2,2(ΩK)↦J0(ΩK) is an unbounded operator with the domain H(ΩK)∩W2,2(ΩK), where P is the projector from L²Ω_K) onto J₀(Ω_K) (Leray’s projector). For given w∈H(ΩK)∩W2,2(ΩK) the operator Δ˜w is defined by

−∫ΩKΔ˜w⋅vdx=∫ΩK(−νΔw+∇p)⋅vdx=−ν∫ΩKΔw⋅vdx=ν∫ΩK∇w⋅∇vdx∀v∈J0∞(ΩK)

(for v∈J0∞(ΩK) holds div v = 0). By density argument,

(3.20) −∫ΩKΔ˜w⋅vdx=−ν∫ΩKΔw⋅vdx∀v∈J0(ΩK).

Hence,

(3.21) −∫ΩK|Δ˜w|2dx=−ν∫ΩKΔw⋅Δ˜wdx.

From (3.20) also follows the estimate

∥Δ˜w∥(H(ΩK))∗≤∥∇w∥L2(ΩK)=∥w∥H(ΩK).

Since H(ΩK)∩W2,2(ΩK) is dense in H (Ω_K), there exists a unique extension of the operator Δ˜ (denoted again by Δ˜ from H(ΩK)∩W2,2(ΩK) to the whole space H (Ω_K). Moreover, the extension Δ˜:H(ΩK)↦(H(ΩK))∗ is a bijection. Δ˜ is called the Stokes operator.

It is known (see, e.g., [21, 22]) that

(i) The Stokes operator has a discrete spectrum:

−Δ˜w=λw,w∈H(ΩK),w≠0;

λi>0,limi→∞λi→+∞.

(ii) The set {w_l} of eigenfunctions of Δ˜ is an orthogonal basis in J₀(Ω_K) and H(ΩK),∥∇wl∥L2(ΩK)=λk, ∥wl∥L2(ΩK)=1. Since ∂Ω_K is smooth, we have wl∈H(ΩK)∩W2,2(ΩK) ^[2]

Relation (3.21) yields

(3.22) ∥Δ˜w∥L2(ΩK)≤c∥Δw∥L2(ΩK)≤c∥∇2w∥L2(ΩK).

From (3.3) follows the estimate

(3.23) ∫Ω0|∇v|2dx+∫ΩK∖Ω0φ−2(x2)|∇v|2dx≤c∫ΩK|Δ˜v|2dx,

and from (3.7) we get the inequality

∥∇2w∥L2(ΩK)≤c∥Δ˜w∥L2(ΩK),

which together with (3.22) implies

(3.24) c1∥∇2w∥L2(ΩK)≤∥Δ˜w∥L2(ΩK)2≤c2∥∇2w∥L2(ΩK).

Note that constants in (3.22),(3.24) are independent of K.

4 Solvability of Navier–Stokes problem

4.1 Construction of the extension of boundary data

Consider problem (1.1)–(1.4). Suppose that a∈L2(0,T;Wloc3/2,2(∂Ω∖{O})),at∈L2(0,T;Wloc1/2,2(∂Ω∖{O})), supp suppa⊂∂Ω0∩∂Ω⊂∂Ω1♯.

First we consider the linear extension operator E in the domain Ω3♯,E:W3/2,2(∂Ω3♯)⟼W2,2(Ω3♯) given by Ea=w(1), where w(1)|∂Ω3♯=a. Since the boundary ∂Ω∩∂Ω3♯ is smooth and supp a⊂∂Ω0∩∂Ω⊂∂Ω1♯, the linear operator E is bounded:

(4.1) ∥Ea∥W2,2(Ω3♯)2=∥w(1)∥W2,2(Ω3♯)2≤c∥a∥W3/2,2(∂Ω)2.

Moreover, w⁽¹⁾ can be constructed so that supp w(1)⊂Ω‾2♯ (see, e.g., [16]).

If a=a(x,t)andat(⋅,t)∈W1/2,2(∂Ω), then, due to the fact that the operator E is linear, we have Eat= wt(1) and

(4.2) ∥Eat∥W1,2(Ω3♯)2≤c∥at∥W1/2,2(∂Ω)2.

Moreover, if a∈L2(0,T;W3/2,2(∂Ω)),at∈L2(0,T;W1/2,2(∂Ω)), then integrating (4.1), (4.2) by t yields

(4.3) ∥w(1)∥L2(0,T;W2,2(Ω3♯))2≤c∥a∥L2(0,T;W3/2,2(∂Ω))2,∥wt(1)∥L2(0,T;W1,2(Ω3♯))2≤c∥at∥L2(0,T;W1/2,2(∂Ω))2.

Let U[J](x1x2λ,x2,t,τ) be the formal asymptotic decomposition of the velocity component near the cusp point O constructed in [1] . Recall that

U[J](x1x2λ,x2,t,τ)=UO,[J](x1x2λ,x2,t)+UB,[J](x1x2λ,x2,tx22λ),

where U^O^,[J] is the outer asymptotic expansion and U^B^,[J] is the boundary layer expansion (see also formulas (1.5) in Introduction). In order to insure the existence of U^[J], the following regularity requirements for the boundary value a are needed:

∂la∂tl∈L2(0,T;W1/2,2(∂Ω)),l=1,2,...,J+1.

It is proved (see inequality (4.15) in [1]) that the vector field U^[J] satisfies the following estimates

(4.4) supt∈[0,T]∥U[J](⋅,y2,t)∥W1,2(Υ)2+∥U[J]∥L2(0,T;W2,2(Υ)2+∥Ut[J]∥L2(0,T;L2(Υ))2≤cφ2(y2)∫0T|||F|||J+12dt,supt∈[0,T]∥U[J](⋅,y2,t)∥W2,2(Υ)2+supt∈[0,T]∥Ut[J]∥L2(Υ)2+∥∇Ut[J]∥L2(0,∞;L2(Υ))2≤cφ2(y2)∫0T|||F|||J+22dt,supt∈[0,T]∥∂U[J](⋅,y2,t)∂y2∥W1,2(Υ)2≤cφ4(y2)∫0T|||F|||J+12dt,

where y1=x1x2λ,y2=x2,φ(y2)=γ0y2λ,λ>1,Υ=(−γ0,γ0),|||F|||J2=∑k=0J|∂kF(t)∂tk|2.

Since W1,2(Υ)⊂C(Υ), we have

(4.5) supt∈(0,T)y1∈Υ(|U[J](y1,y2,t)|2+|∂U[J](y1,y2,t)∂y1|2)≤csupt∈(0,T)∥U[J](⋅,y2,t)∥W2,2(Υ)2≤cφ2(y2)∫0T|||F|||J+22dt,supt∈(0,T)y1∈Υ|∂U[J](y1,y2,t)∂y2|2dt≤csupt∈(0,T)∥∂U[J](⋅,y2,t)∂y2∥W1,2(Υ)2≤cφ4(y2)∫0T|||F|||J+12dt.

Passing to coordinates x we obtain

(4.6) supt∈(0,T)x1∈(−φ(x2),φ(x2))|U[J](x1,x2,t)|2dt≤cφ2(x2)∫0T|||F|||J+12dt,supt∈(0,T)x1∈(−φ(x2),φ(x2))|∇xU[J](x1,x2,t)|2≤cφ4(x2)∫0T|||F|||J+12dt.

Notice that F(t)=∫∂Ωa⋅ndS, and hence,

(4.7) ∫0T|||F|||J2dt≤c∫0T|||a|||J2dt,supt∈(0,T)|||F|||J2dt≤c∫0T|||a|||J+12dt,

where |||a|||J2=∑k=0J∥∂ka(⋅,t)∂tk∥W1/2,2(∂Ω)2.

Consider the function B=w(1)+ζU[J], where ζ=ζ(x2) is a smooth cut-off function equal to one in Ω∖Ω2♯ and equal to zero in Ω1♯. Obviously, B|_∂Ω = a, however, B is not solenoidal, divB=divw(1)+∇ζ⋅U[J]:=h. Notice that

∫Ω2♯hdx=∫∂Ω2♯(w(1)+ζU[J])⋅ndS=∫∂Ω0∩∂Ωa⋅ndS+∫∂Ω2♯∖∂ΩU[J]⋅ndS=F(t)−F(t)=0.

Since supp h⊂Ω‾2♯ and the boundary ∂Ω3♯∩∂Ω is smooth, there exist a function w(2)∈W2,2(Ω3♯) such that supp suppw(2)⊂Ω‾3♯,w(2)=0 in the neighbourhood of ∂Ω3♯∖∂Ω and

(4.8) divw(2)=hinΩ3♯,w|∂Ω3♯(2)=0.

Moreover,

(4.9) ∥w(2)(⋅,t)∥W2,2(Ω3♯)2≤c∥h(⋅,t)∥W1,2(Ω3♯)2≤c(∥w(1)(⋅,t)∥W2,2(Ω3♯)2+∥U[J](⋅,t)∥W1,2(Ω3♯)2)

(see [23]). By construction in [23], it follows that the operator 𝓓 of problem (4.8), D:w(2)(⋅,t)∈W2,2(Ω3♯)⟼ div w(2)(⋅,t)=h(⋅,t)∈W1,2(Ω3♯) is linear and the inverse operator 𝓓⁻¹ defined on functions h∈W1,2(Ω3♯), satisfying the condition ∫Ω3♯hdx=0, is bounded. Moreover, equations (4.8) can be differentiated with respect

to t:

divwt(2)=htinΩ3♯,wt|∂Ω3♯(2)=0,

and

(4.10) ∥wt(2)(⋅,t)∥W1,2(Ω3♯)2≤c∥ht(⋅,t)∥L2(Ω3♯)2≤c(∥wt(1)(⋅,t)∥W1,2(Ω3♯)2+∥Ut[J](⋅,t)∥L2(Ω3♯)2).

Integrating inequalities (4.9), (4.10) with respect to t and using estimates (4.3), (4.4) and (4.7) we obtain

(4.11) ∥w(2)∥L2(0,T;W2,2(Ω3♯))2≤c∫0T(∥a(⋅,t)∥W3/2,2(∂Ω)2+|||a(⋅,t)|||J+12+|||F(t)|||J+12)dt≤c∫0T(∥a(⋅,t)∥W3/2,2(∂Ω)2+|||a(⋅,t)|||J+12)dt=c∫0T〈a〉J+12dt,∥wt(2)∥L2(0,T;W1,2(Ω3♯))2≤c∫0T(∥at(⋅,t)∥W1/2,2(∂Ω)2+|||F(t)|||J+12)dt≤∫0T|||a(⋅,t)|||J+12dt,

where 〈a〉J+12=∥a(⋅,t)∥W3/2,2(∂Ω)2+|||a(⋅,t)|||J+1.

Define

W=w(1)+w(2),V=W+ζU[J],

where ζ is a smooth cut-off function defined above. By construction div V = 0, V|_∂Ω = a and V = U^[J] for x∈Ω∖Ω3♯. Therefore, for x∈Ω∖Ω3♯ the function V satisfies estimates (4.4), (4.6), while for x∈Ω3♯ from (4.11) and (4.4) it follows that

(4.12) ∫0T∥V(⋅,t)∥W2,2(Ω3♯)2dt+∫0T∥Vt(⋅,t)∥L2(Ω3♯)2dt≤c∫0T(∥a∥W3/2,2(∂Ω)2+|||a|||J+12+|||F|||J+12)dt≤c∫0T〈a〉J+12dt.

We look for the solution (u, p) of problem (1.1) in the form

u=v+V,p=q+ζP[J].

Then for (v, q) we obtain the following problem

(4.13) vt−νΔv+(v⋅∇)v+(V⋅∇)v+(v⋅∇)V+∇q=fˆ,divv=0,v|∂Ω∖O=0,v(x,0)=uˆ0(x),

where fˆ=f−f1,f1=Vt−νΔV+(V⋅∇)(V)+∇(ζP[J]),uˆ0=b−W|t=0 (since U[J](x,0)=0). Recall that the number J was chosen such that

Ut[J]−νΔU[J]+(U[J]⋅∇)U[J]+∇P[J]=H(J)∈L2(0,T;L2(GH)).

(see [1]). Therefore, taking into account that W has compact support Ω‾3♯, we conclude

fˆ∈L2(0,T;L2(Ω)),uˆ0∈W∘ 1,2(Ω).

Moreover, using results obtained in [1] we get (see estimate (4.19) in [1])

(4.14) ∥fˆ∥L2(0,T;L2(Ω))2≤c(f∥L2(0,T;L2(Ω))2+∫0T〈a〉J+12dt.),∥uˆ0∥W1,2(Ω)2≤c(∥b∥W1,2(Ω)2+∫0T〈a〉J+12dt).

In the next subsections we construct the sequence of weak solutions v_K to the Navier–Stokes equations in regular domains Ω_K and prove the uniform (with respect to K) estimates for them. The solution of problem (4.13) is then found as a limit of {v_K}.

4.2 Existence of the solution in Ω_K

Consider in Ω_K the following boundary value problem

(4.15) vKt−νΔvK+(vK⋅∇)vK+(V⋅∇)vK+(vK⋅∇)V+∇qK=fˆ,divvK=0,vK|∂ΩK=0,vK(x,0)=uˆK0(x),

where uˆK0∈W∘ 1,2(ΩK),divuˆK0=0and∥uˆK0−uˆ0∥W1,2(Ω)→0 as K → +∞ (here we suppose that uˆK0 is extended by zero into Ω \ Ω_K).

In this subsection we omit the subscript K in notation of the solution v_K.

By a weak solution of problem (4.15) we mean the function v ∈ L²(0, T; H(Ω_K)) with vt,∇2v∈ L2(0,T;L2(ΩK)) satisfying the initial condition v(x,0)=uˆK0(x), and for all t ∈ [0, T] satisfying the integral identity

(4.16) ∫0T∫ΩK(vt⋅η+ν∇v⋅∇η−((v+V)⋅∇)η⋅v−(v⋅∇)η⋅V)dxdt=∫0T∫ΩKfˆ⋅ηdxdt

for any test function η∈L2(0,T;H(ΩK)) with ηt∈L2(0,T;L2(ΩK)).

Lemma 4.1

Let uˆ0∈W∘ 1,2(Ω). There exists a sequence uˆK0∈W∘ 1,2(ΩK) such that div uˆK0=0andlimK→∞∥uˆK0 −uˆ0∥W1,2(Ω)=0. Moreover there hods the estimate

(4.17) ∥uˆK0∥W1,2(Ω)2≤c(∥b∥W1,2(Ω)2+∫0T〈a〉J+12dt)

with the constant c independent of K.

Proof

Let χ_K(x²) be a smooth cut-off function such that χK(x2)=0forx2≥hK−1,χK(x2)=1forx2≤hK−2 and |∇χ(x2)|≤cφ(hK−1). Consider the sequence of functions uˆK0=χK(x2)uˆ0+wˆK, where wˆK is a solution of the problem

divwˆK=−∇χK⋅uˆ0,x∈ωK−1,wˆK=0,x∈∂ωK−1.

Obviously, ∫ωK−1∇χK⋅uˆ0dx=0. Therefore, by Lemma 2.4, there exists wˆK∈W∘ 1,2(ωK−1) satisfying the estimate

(4.18) ∥∇wˆK∥L2(ωK−1)≤c∥∇χK⋅uˆ0∥L2(ωK−1)≤c∥φ−1uˆ0∥L2(ωK−1)≤c∥∇uˆ0∥L2(ωK−1)

with the constant c independent of K. By construction div uˆK0=0 and

∥uˆK0−uˆ0∥W1,2(Ω)≤∥(1−χK)uˆ0∥W1,2(ωK−1)+∥wˆK∥W1,2(ωK−1)≤c∥uˆ0∥W1,2(ωK−1).

Since the right-hand side of this inequality tend to zero as K → ∞, we get limK→∞∥uˆK0−uˆ0∥W1,2(Ω)=0. Estimate (4.17) follows from (4.14).

Theorem 4.1

Suppose that f ∈ L²(0, T; Ω), b ∈ W^1,2(Ω), the boundary value a has the finite norm ∫0T〈a〉J+22dt and let there hold the compatibility conditions div b = 0, b|_∂Ω = a(x, 0). There exists a positive constant κ₀ such that if the flux F (t) satisfies the inequality ∫0T|||F|||J+22dt≤κ0, then problem (4.15) admits at least one weak solution v. The following estimates

(4.19) supt∈[0,T]∥v(⋅,t)∥L2(ΩK)2+∫0T∥∇v(⋅,t)∥L2(ΩK)2dt≤cB1

(4.20) supt∈[0,T]∥∇v(⋅,t)∥L2(ΩK)2≤ceB2(A0+A1+B1):=cB3;

(4.21) ∫0T∥∇2v(⋅,t)∥L2(ΩK)2dt≤cB4

(4.22) ∫0T∥vt(⋅,t)∥L2(ΩK)2dt≤cB5

hold. The constants in estimates (4.19)–(4.22) are independent of K and

A0=∥b∥W1,2(Ω)2+∥f∥L2(0,T;L2(Ω))2,A1=∫0T〈a〉J+12dt,B1=(1+ec4A1A1)(A0+A1),B2=c9A1+c10B12,B3=eB2(A0+A1+B1),B4=A0+A1+A1B3+B3B12+B1,B5=A0+A1+(1+A1)B4+(1+B1B3)B1+B3A1.

The constants c₄, c₉ and c₁₀ are defined in the proof of the theorem.

Proof

We follow the scheme of O.A. Ladyzhenskaya book [22] (see also [21, 24]) where the solvability of problem (4.15) is proved by the Galerkin method. Let {ψl}l=1∞ be an orthogonal basis in the space H (Ω_K). Consider Galerkin approximations v(N)(x,t)=∑l=1Nγl(N)(t)ψl(x) of the solution v of problem (4.15) which are defined by the following system of ordinary differential equations (with respect to functions γl(N)(t),l=1,…,N):

(4.23) ∫ΩK(vt(N)⋅ψl+ν∇v(N)⋅∇ψl−((v(N)+V)⋅∇)ψl⋅v(N)−(v(N)⋅∇)ψl⋅V)dx=−∫ΩKfˆ⋅ψldx,l=1,…,N,γl(N)(0)=αl,l=1,…,N,

where α_l are the coefficients of the initial function uˆK0 in the basis {ψk}k=1∞,i.e.,uˆK0=∑l=1∞αlψl(x).

Multiplying (4.23) by γl(N)(t) and summing by l from 1 to N we obtain

12ddt∫ΩK|v(N)|2dx+ν∫ΩK|∇v(N)|2dx=∫ΩK(v(N)⋅∇)v(N)⋅Vdx+∫ΩKfˆ⋅v(N)dx.

Consider the integral

|∫ΩK(v(N)⋅∇)v(N)⋅Vdx|≤|∫Ω3♯(v(N)⋅∇)V⋅v(N)dx|+|∫ΩK∖Ω3♯(v(N)⋅∇)V⋅v(N)dx|=I1+I2.

For I₁ we have the estimate

|I1|≤∥∇V∥L2(Ω3♯)∥v(N)∥L4(Ω3♯)2≤c∥∇V∥L2(Ω3♯)∥v(N)∥L2(Ω3♯)∥∇v(N)∥L2(Ω3♯)≤ν4∫ΩK|∇v(N)|2dx+c∥∇V∥L2(Ω3♯)2∫ΩK|v(N)|2dx.

Consider I₂. By Poincaré inequality (2.4) and (4.6),

|∫ΩK∖Ω3♯(v(N)⋅∇)V⋅v(N)dx|≤∫ΩK∖Ω3♯|v(N)|2|∇U[J]|dx≤c(∫0T|||F|||J+22dt)1/2∫ΩK∖Ω3♯|v(N)(x)|2φ2(x2)dx≤c1(∫0T|||F|||J+22dt)1/2∫ΩK|∇v(N)|2dx.

Further,

|∫ΩKfˆ⋅v(N)dx|≤c(ε)(∫Ω3♯|fˆ|2dx+∫ΩK∖Ω3♯φ2(x2)|fˆ|2dx)+ε(∫ΩK∖Ω3♯|v(N)|2φ2(x2)dx+∫Ω3♯|v(N)|2dx)≤c(ε)∫ΩK|fˆ|2dx+c2ε∫ΩK|∇v(N)|2dx.

Thus, for F(t) such that (∫0T|||F|||J+22dt)1/2≤ν4c1andε=ν4c2 we obtain

(4.24) 12ddt∫ΩK|v(N)|2dx+ν4∫ΩK|∇v(N)|2dx≤c3∥∇V∥L2(Ω3♯)2∫ΩK|v(N)|2dx+c∫ΩK|fˆ|2dx.

By Gronwall’s inequality, (4.24) yields

Inequalities (4.24), (4.25) imply

where

Aˆ=∫ΩK|uˆK0(x)|2dx+∫0T∥fˆ(⋅,t)∥L2(Ω)2dt≤c(∥b∥W1,2(Ω)2+∥f∥L2(0,T;L2(Ω))2)+c∫0T〈a〉J+12dt:=c(A0+A1)

(see (4.14), (4.17)). By (4.12),

(4.26) ∫0T∥V(⋅,t)∥W2,2(Ω3♯)2dt≤c∫0T〈a〉J+12dt.

Therefore, using (4.17) we obtain

(4.27) supt∈[0,T]∥v(N)(⋅,t)∥L2(ΩK)2+ν∫0T∫ΩK|∇v(N)(x,t)|2dxdt≤c5(1+ec4A1A1)(A0+A1):=c5B1.

Estimate (4.27) guaranties that the Cauchy problem (4.23) admits a unique solution for each fixed N. Now we derive a number of a priori estimates for Galerkin approximations v^(N). Estimate (4.27) is valid for Galerkin approximations constructed using an arbitrary orthogonal basis. In order to estimate the higher derivatives of v^(N), as a basis we shall use the eigenfunctions of the Stokes operator.

Taking in (4.23) ψ_l = w_l, where w_l are eigenfunctions of the Stokes operator, i.e., −Δ˜wl=λlwl, multiplying the obtained relations by λlγl(N)(t) and summing by l from 1 to N we obtain

∫ΩK(vt(N)⋅Δ˜v(N)−νΔv(N)⋅Δ˜v(N)+((v(N)+V)⋅∇)v(N)⋅Δ˜v(N)+(v(N)⋅∇)V⋅Δ˜v(N))dx=∫ΩKfˆ⋅Δ˜vdx.

This is equivalent to (see the properties of the Stokes operator)

(4.28) 12ddt∫ΩK|∇v(N)|2dx+∫ΩK|Δ˜v(N)|2dx=−∫ΩK((v(N)+V)⋅∇)v(N)⋅Δ˜v(N)dx−∫ΩK(v(N)⋅∇)V⋅Δ˜v(N)dx+∫ΩKfˆ⋅Δ˜v(N)dx=∑i=13Ji.

Let us estimate the right-hand side of (4.28). By Young’s inequality,

(4.29) |J3|≤ε∫ΩK|Δ˜v(N)|2dx+cε∫ΩK|fˆ|2dx.

Further, by (4.6), (2.4) and (3.23),

(4.30) |J2|≤{∥∇V∥L4(Ω3♯)∥v(N)∥L4(Ω3♯)+(∫ΩK∖Ω3♯|v(N)|2|∇U[J]|2dx)1/2}×(∫ΩK|Δ˜v(N)|2dx)1/2≤ε∫ΩK|Δ˜v(N)|2dx+cε∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+(∫ΩK∖Ω3♯|v(N)|2|∇U(J)|2dx)1/2(∫ΩK|Δ˜v(N)|2dx)1/2≤ε∫ΩK|Δ˜v(N)|2dx+cε∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+(∫0T|||F|||J+22dt)1/2(∫ΩK∖Ω3♯|v(N)|2φ4(x2)dx)1/2(∫ΩK|Δ˜v(N)|2dx)1/2≤ε∫ΩK|Δ˜v(N)|2dx+cε∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+(∫0T|||F|||J+22dt)1/2(∫ΩK∖Ω3♯|∇v(N)|2φ2(x2)dx)1/2(∫ΩK|Δ˜v(N)|2dx)1/2≤ε∫ΩK|Δ˜v(N)|2dx+cε∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+(∫0T|||F|||J+22dt)1/2∫ΩK|Δ˜v(N)|2dx.

Similarly (J₁ = J₁₁ + J₁₂) we obtain the estimates

(4.31) |J11|=|∫ΩK(V⋅∇)v(N)⋅Δ˜v(N)dx|≤∥V∥L4(Ω3♯)∥∇v(N)∥L4(Ω3♯)(∫ΩK|Δ˜v(N)|2dx)1/2+c(∫0T|||F|||J+22dt)1/2(∫ΩK∖Ω3♯|∇v(N)|2φ2(x2)dx)1/2(∫ΩK|Δ˜v(N)|2dx)1/2≤c∥V∥W1,2(Ω3♯)∥v(N)∥W2,2(Ω3♯)(∫ΩK|Δ˜v(N)|2dx)1/2+c(∫0T|||F|||J+22dt)1/2(∫ΩK∖Ω3♯|∇v(N)|2φ2(x2)dx)1/2(∫ΩK|Δ˜v(N)|2dx)1/2≤c(ε+(∫0T|||F|||J+22dt)1/2)|∫ΩK|Δ˜v(N)|2dx+cε∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2,

and

(4.32) |J12|=|∫ΩK(v(N)⋅∇)v(N)⋅Δ˜v(N)dx|≤ε∫ΩK|Δ˜v(N)|2dx+cε(∥v(N)∥L4(Ω3♯)2∥∇v(N)∥L4(Ω3♯)2+∑l=1K−1∥v(N)∥L4(ωl)2∥∇v(N)∥L4(ωl)2+∥v(N)∥L4(ωˆK)2∥∇v(N)∥L4(ωˆK)2).

Applying inequalities (2.6) and (2.5) we get

(4.33) ∥v(N)∥L4(Ω3♯)2∥∇v(N)∥L4(Ω3♯)2≤c∥v(N)∥L2(Ω3♯)∥∇v(N)∥L2(Ω3♯)2(∥∇v(N)∥L2(Ω3♯)2+∥∇2v(N)∥L2(Ω3♯)2)1/2≤cε(∥∇v(N)∥L2(Ω3♯)2+∥v(N)∥L2(Ω3♯)2∥∇v(N)∥L2(Ω3♯)4)+ε∥∇2v(N)∥L2(Ω3♯)2;

(4.34) ∥v(N)∥L4(ωl)2∥∇v(N)∥L4(ωl)2≤cφ−1(hl−1)∥v(N)∥L2(ωl)∥∇v(N)∥L2(ωl)2(∥∇v(N)∥L2(ωl)2+φ2(hl−1)∥∇2v(N)∥L2(ωl)2)1/2≤cε∥v(N)∥L2(ωl)2∥∇v(N)∥L2(ωl)4+ε(∥∇2v(N)∥L2(ωl)2+φ−2(hl−1)∥∇v(N)∥2);

(4.35) ∥v(N)∥L4(ωˆK)2∥∇v(N)∥L4(ωˆK)2≤cε∥v(N)∥L2(ωˆK)2∥∇v(N)∥L2(ωˆK)4+ε(∥∇2v(N)∥L2(ωˆK)2+φ−2(hK−1)∥∇v(N)∥L2(ωˆK)2).

Thus, (4.32)–(4.35) and (3.23), (3.24) imply

(4.36) |J12|≤cε(∫ΩK|Δ˜v(N)|2dx+∫ΩK|∇2v(N)|2dx+∫ΩK∖Ω3♯|∇v(N)|2φ2(x2)dx)+c(∥∇v(N)∥L2(Ω3♯)2+∥v(N)∥L2(Ω3♯)2∥∇v(N)∥L2(Ω3♯)4)+c∑l=1K−1∥v(N)∥L2(ωl)2∥∇v(N)∥L2(ωl)4+c∥v(N)∥L2(ωˆK)2∥∇v(N)∥L2(ωˆK)4≤cε∫ΩK|Δ˜v(N)|2dx+c∥∇v(N)∥L2(ΩK)2+∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)2×(∫Ω3♯|∇v(N)|2dx+∑l=1K−1∫ωl|∇v(N)|2dx+∫ωˆK|∇v(N)|2dx)≤cε∫ΩK|Δ˜v(N)|2dx+c∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)4+c∥∇v(N)∥L2(ΩK)2.

Substituting (4.29)–(4.31), (4.36) into (4.28) yields

(4.37) 12ddt∫ΩK|∇v(N)|2dx+∫ΩK|Δ˜v(N)|2dx≤c6(ε+(∫0T|||F|||J+22dt)1/2)∫ΩK|Δ˜u(N)|2dx+c∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+c∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)4+c∥∇v(N)∥L2(ΩK)2+c∫ΩK|fˆ|2dx.

Taking in (4.37) ε=14c6 and assuming that (∫0T|||F|||J+22dt)1/2≤14c6 we derive

(4.38) ddt∫ΩK|∇v(N)|2dx+∫ΩK|Δ˜v(N)|2dx≤c7∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2+c8(∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)4+∥∇v(N)∥L2(ΩK)2)+c9∫ΩK|fˆ|2dx

Denoting

Y(t)=∫ΩK|∇v(N)(x,t)|2dx,A(t)=c9∫ΩK|fˆ|2dx+c8∥∇v(N)∥L2(ΩK)2,B(t)=c7∥V∥W2,2(Ω3♯)2+c8∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)2,

we rewrite (4.38) as

Y′(t)≤B(t)Y(t)+A(t).

By Gronwall’s lemma,

(4.39) Y(t)≤e∫0tB(τ)dτ(Y(0)+∫0te−∫0sB(τ)dτA(s)ds)

≤e∫0TB(τ)dτ(Y(0)+∫0TA(s)ds).

Estimates (4.26), (4.27) yield

∫0TB(t)dt=c7∫0T∥V∥W2,2(Ω3♯)2dt+c8∫0T∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)2dt≤c∫0T∥V∥W2,2(Ω3♯)2dt+csupt∈[0,T]∥v(N)(⋅,t)∥L2(ΩK)2∫0T∫ΩK|∇v(N)|2dxdt≤c9A1+c10B12:=B2.

Therefore, from (4.39) and (4.27) we have

(4.40) supt∈(0,T)]∥∇v(N)(⋅,t)∥L2(ΩK)2≤ceB2(∥∇uˆk0∥L2(ΩK)2+∥∇v(N)∥L2(0,T;L2(ΩK))2+∥fˆ∥L2(0,T;L2(Ω))2)≤c11eB2(A0+A1+B1)=c11B3.

Substituting (4.40) into (4.38) and integrating over t yield

(4.41) ∫0T∫ΩK|Δ˜v(N)|2dxdt≤∫ΩK|∇uˆ0k|2dx+c∫0T∫Ω|fˆ|2dxdt+c∫0T∥V∥W2,2(Ω3♯)2∥∇v(N)∥L2(ΩK)2dt+∫0T∥∇v(N)∥L2(ΩK)2dt+c∫0T∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)4dt≤c(A0+A1)+csupt∈(0,T)∥∇v(N)∥L2(ΩK)2∫0T∥V∥W2,2(Ω3♯)2dt+csupt∈(0,T)∥v(N)∥L2(ΩK)2supt∈(0,T)∥∇v(N)∥L2(ΩK)2∫0T∥∇v(N)∥L2(ΩK)2dt+c∫0T∥∇v(N)∥L2(ΩK)dt2≤c12[A0+A1+B3A1+B3B12+B1]:=c12B4.

Let us estimate the norm of ut(N). Multiplying (4.23) by ddtγl(N)(t) and summing by l from l = 1 to l = N we obtain

(4.42) ∫ΩK|vt(N)|2dx+ν2ddt∫ΩK|∇v(N)|2dx=−∫ΩK((v(N)+V)⋅∇)v(N)⋅vt(N)dx−∫ΩK(v(N)⋅∇)V⋅vt(N)dx+∫ΩKf⋅vt(N)dx≤12∫ΩK|vt(N)|2dx+c∫Ω|fˆ|2dx+c∫ΩK(|v(N)|2+|V|2)|∇v(N)|2dx+c∫ΩK|v(N)|2|∇V|2dx.

Let us estimate the last two terms in the right-hand side of (4.42). By the same argument as before,

∫ΩK|v(N)|2|∇V|2dx≤c((∫Ω3♯|v(N)|4dx)1/2(∫Ω3♯|∇V|4dx)1/2+∫0T|||F|||J+22dt∫ΩK∖Ω3♯|v(N)|2φ4(x2)dx)≤c(∥V∥W2,2(Ω3♯)2∫ΩK|∇v(N)|2dx+∫0T|||F|||J+22dt∫ΩK|Δ˜v(N)|2dx);

∫ΩK|V|2|∇v(N)|2dx≤c(supx∈Ω3♯|V(x,t)|2∫Ω3♯|∇v(N)|2dx+∫0T|||F|||J+22dt∫ΩK∖Ω3♯|∇v(N)|2φ2(x2)dx)≤c(∥V∥W2,2(Ω3♯)2∫ΩK|∇v(N)|2dx+∫0T|||F|||J+22dt∫ΩK|Δ˜v(N)|2dx).

To estimate the integral ∫ΩK|v(N)|2|∇v(N)|2dx, we apply inequalities (4.33)-(4.34) and argue as in the proof of (4.36). In virtue of (2.5) applied to ∇v^(N), we obtain

∫ΩK|v(N)|2|∇v(N)|2dx≤c∫ΩK|Δ˜v(N)|2dx+c∥∇v(N)∥L2(ΩK)2+c∥v(N)∥L2(ΩK)2∥∇v(N)∥L2(ΩK)4≤c∫ΩK|Δ˜v(N)|2dx+c∥∇v(N)∥L2(ΩK)2+csupt∈[0,T]∥v(N)(⋅,t)∥L2(ΩK)2supt∈[0,T]∥∇v(N)(⋅,t)∥L2(ΩK)2)∥∇v(N)|L2(ΩK)2≤c∫ΩK|Δ˜v(N)|2dx+c(1+B1B3)∫ΩK|∇v(N)|2dx.

Substituting the obtained inequalities into (4.42) yields

12∫ΩK|vt(N)|2dx+ν2ddt∫ΩK|∇v(N)|2dx≤c∫Ω|fˆ|2dx+c(1+∫0T|||F|||J+22dt)∫ΩK|Δ˜v(N)|2dx+c(1+B1B3)∫ΩK|∇v(N)|2dx+c∥V∥W2,2(Ω3♯)2∫ΩK|∇v(N)|2dx.

Integrating this inequality over [0, T] and using estimates (4.27), (4.41) we derive

(4.43) ∫0T∫ΩK|vt(N)|2dxdt≤c(∫ΩK|∇uˆ0k|2dx+c∫Ω|fˆ|2dx)+c(1+supt∈(0,T)∫0T|||F|||J+22dt)∫0T∫ΩK|Δ˜v(N)|2dxdt+c(1+B1B3)∫0T∫ΩK|∇v(N)|2dxdt+supt∈(0,T)∥∇v(N)(⋅,t)∥L2(ΩK)2∫0T∥V∥W2,2(Ω3♯)2dt≤c13[A0+A1+(1+A1)B4+(1+B1B3)B1+B3A1]:=c13B5.

Estimates (4.27), (4.41) and (4.43) ensure that there exist a subsequence {v(Nl)} such that {v(Nl)},{∇v(Nl)}, {∇2v(Nl)},{vt(Nl)} are weakly convergent in L²(0, T; L²(Ω_K)). The limit v of this subsequence satisfies the integral identity (4.16) and thus, is a weak solution of (4.15). This part of the proof is standard (see [21, 22, 24]) and we omit the details. Remind that in this section we have omitted the subscript K for notation of the solution and Galerkin approximations, so v(N)=vK(N)andv=vK.

4.3 Existence and uniqueness of the solution to problem (4.13)

By a weak solution of problem (4.13) in the cusp domain Ω we mean the function v ∈ L²(0, T; H(Ω)) with v_t ,∇²v ∈ L²(0, T; L²(Ω)) satisfying the initial condition v(x,0)=uˆ0(x), and for all t ∈ [0, T] satisfying the integral identity

(4.44) ∫0t∫Ω(vt⋅η+ν∇v⋅∇η−((v+V)⋅∇)η⋅v−(v⋅∇)η⋅V)dxdt=∫0t∫Ωfˆ⋅ηdxdt

for any test function η ∈ L²(0, T; H(Ω)), η_t ∈ L²(0, T; L²(Ω)) having compact support in Ωˉ∖{O} (i.e., the support of η is separated from the cusp point O)

Theorem 4.2

Assume that the conditions of Theorem 4.1 are valid. There exists a number κ₁ > 0 such that if ∫0T|||F|||J+22dt≤κ1, then there exists a unique weak solution v of problem (4.13).

Proof

Let ∫0T|||F|||J+22dt≤κ0,

where κ₀ is a number from Theorem 4.1. Then, due to estimates (4.19)–(4.22) for

v_K,we can extract a subsequence {vKl} such that {vKl},{∇vKl},{∇2vKl}and{vKl,t} are weakly convergent in L²(0, T; L²(Ω)) as Kl→∞ Taking in (4.16) a test function η with the compact support and passing K_l to infinity we obtain for the limit v integral identity (4.44). Moreover, obviously v(x,0)=uˆ0(x) and thus, v is a weak solution of problem (4.13). For v remain valid estimates (4.19)–(4.22). Let us prove the uniqueness. Let u₁ and u₂ be two solutions of problem (1.1) having the same representations: u1=V+v1andu2=V+v2, where v1,v2 are solutions of problem (4.13). The difference u1−u2=v1−v2:= v∈W∘ 1,2(Ω) satisfies zero initial condition v(x, 0) = 0 and the integral identity

(4.45) ∫0t∫Ω(vt⋅η+ν∇v⋅∇η)dxdt−∫0t∫Ω(V⋅∇)η⋅vdxdt−∫0t∫Ω(v⋅∇)η⋅Vdxdt−∫0t∫Ω(v⋅∇)η⋅v1dxdt=0.

Let us take in (4.45) η = χ_K(x₂)v + w_K, where χ_K is defined in the proof of Lemma 4.1 and w_K is a solution of the problem

divwK=−∇χK⋅v,x∈ωK−1,wK=0,x∈∂ωK−1,

satisfying the estimate

(4.46) ∥∇wK∥L2(ωK−1)≤c∥∇χK⋅v∥L2(ωK−1)≤c∥φ−1v∥L2(ωK−1)≤c∥∇v∥L2(ωK−1).

This gives

(4.47) 12∫0tddt∫ΩχK|v|2dxdt+ν∫0t∫ΩχK|∇v|2dxdt=−∫0t∫Ωvt⋅wKdxdt−ν∫0t∫Ω∇v⋅∇χK⋅vdxdt−ν∫0t∫Ω∇v⋅∇wKdxdt+∫0t∫Ω(V⋅∇)(χKv+wK)⋅vdxdt+∫0t∫Ω(v⋅∇)(χKv+wK)⋅Vdxdt+∫0t∫Ω(v⋅∇)(χKv+wK)⋅v1dxdt:=∑i=16Ji.

Let us estimate the right-hand side of (4.47). Using the properties of χ_K and w_K we obtain

(4.48) |J1|≤∫0t∥vt∥L2(ωK−1)∥wK∥L2(ωK−1)dt≤c∫0t(∥vt∥L2(ωK−1)2+∥∇v∥L2(ωK−1)2)dt;|J2|+|J3|≤∫0t∥∇v∥L2(ωK−1)∥φ−1v∥L2(ωK−1)dt+∫0t∥∇v∥L2(ωK−1)∥∇wK∥L2(ωK−1)dt≤c∫0t∥∇v∥L2(ωK−1)2dt;

Integrating by parts yields

∫0t∫Ω(V⋅∇)(χKv)⋅vdxdt=12∫0t∫ωK−1(V⋅∇)χK|v|2dxdt.

Therefore, by (4.6), (4.46),

(4.49) |J4|≤(∫0T|||F|||J+22dt)1/2∫0t(∥φ−1v∥L2(ωK−1)2+∥φ−1v∥L2(ωK−1)∥∇wK∥L2(ωK−1))dt≤(∫0T|||F|||J+22dt)1/2∫0t∥∇v∥L2(ωK−1)2dt.

Moreover, using (4.12), (4.6), (2.5) we get

(4.50) |J5|≤|∫0t∫Ω(v⋅∇)V⋅(χKv+wK)dxdt|≤|∫0t∫Ω3♯(v⋅∇)V⋅vdxdt|+c(∫0T|||F|||J+22dt)1/2∫0t∫Ω∖Ω3♯χKφ−2|v|2dxdt+|∫0t∫ωK−1(v⋅∇)V⋅wKdxdt|≤c(∫0T|||F|||J+22dt)1/2∫0t∫Ω∖Ω3♯χK|∇v|2dxdt+c∫0t((∫0T|||F|||J+22dt)1/2∥φ−1v∥L2(ωK−1)∥φ−1wK∥L4(ωK−1)+∥∇V∥L2(Ω3♯)∥v∥L4(Ω3♯)2)dt≤c(∫0T|||F|||J+22dt)J+11/2∫0t∫Ω∖Ω3♯χK|∇v|2dxdt+c∫0t∥∇V∥L2(Ω3♯)∥v∥L2(Ω3♯)∥∇v∥L2(Ω3♯)dt+c∫0t∥∇v∥L2(ωK−1)2dt≤c((∫0T|||F|||J+22dt)1/2+ε)∫0t∫ΩχK|∇v|2dxdt+cε∫0t∥∇v∥L2(ωK−1)2dt+c∫0t∥∇V∥L2(Ω3♯)2∥χKv∥L2(Ω)2dt

and

(4.51) |J6|≤|∫0t∫ΩχK(v⋅∇)v⋅v1dxdt|+|∫0t∫ωK−1(v⋅∇)χK(v⋅v1)dxdt|+|∫0t∫ωK−1(v⋅∇)wK⋅v1dxdt|≤ε∫0t∫ΩχK|∇v|2dxdt+cε∫0t∥χKv∥L4(Ω)2∥v1∥L4(Ω)2dt+c∫0t∥φ−1v∥L2(ωK−1)∥v∥L4(ωK−1)∥v1∥L4(ωK−1)dt+c∫0t∥∇wK∥L2(ωK−1)∥v∥L4(ωK−1)∥v1∥L4(ωK−1)dt≤ε∫0t∫ΩχK|∇v|2dxdt+c∫0t∥χKv∥L2(Ω)∥∇(χKv)∥L2(Ω)∥∇v1∥L2(Ω)2dt+c∫0t∥∇v1∥L2(ωK−1)∥∇v∥L2(ωK−1)2dt≤cε∫0t∫ΩχK|∇v|2dxdt+cε∫0t∥∇v1∥L2(Ω)4∥χKv∥L2(Ω)2dt+c∫0t∥∇v1∥L2(ωK−1)∥∇v∥L2(ωK−1)2dt.

Substituting estimates (4.48)–(4.51) into (4.47) we obtain

(4.52) 12∫0tddt∫ΩχK|v|2dxdt+ν∫0t∫ΩχK|∇v|2dxdt≤c14((∫0T|||F|||J+22dt)1/2+ε)∫0t∫ΩχK|∇v|2dxdt+c∫0t(∥∇V∥L2(Ω3♯)2+∥∇v1∥L2(Ω)4)∥χKv∥L2(Ω)2dt+c∫0t[∥vt∥L2(ωK−1)+∥∇v1∥L2(ωK−1)∥∇v∥L2(ωK−1)2]dt.

Taking ε=ν4c14 in (4.52) and assuming that c14(∫0T|||F|||J+22dt)1/2≤ν4 we get

(4.53) ∫ΩχK|v(x,t)|2dx≤c∫0t(∥∇V∥L2(Ω3♯)2+∥∇v1∥L2(Ω)4)∫ΩχK|v(x,t)|2dxdt+c∫0t[∥vt∥L2(ωK−1)+∥∇v1∥L2(ωK−1)∥∇v∥L2(ωK−1)2]dt.

Introduce the notation:

Z(t)=∫ΩχK|v(x,t)|2dx,β(t)=(∥∇V(⋅,t)∥L2(Ω3♯)2+∥∇v1(⋅,t)∥L2(Ω)4),α(t)=∫0t[∥vt∥L2(ωK−1)+∥∇v1∥L2(ωK−1)∥∇v∥L2(ωK−1)2]dt.

Then (4.53) can be written as

Z(t)≤c15α(t)+c16∫0tβ(τ)Z(τ)dτ.

By Gronwall lemma, the last inequality yields

(4.54) Z(t)≤c15α(t)exp(c16∫0tβ(τ)dτ).

Estimates (4.19), (4.20) for the solution v₁ and estimate (4.26) imply (see Theorem 4.1)

∫0T∥∇v1∥L2(Ω)2dt≤cB1,supt∈(0,T)∥∇v1∥L2(Ω)2≤cB3,∫0T∥∇V∥L2(Ω3♯)2dt≤cA1.

This together with (4.54) yields

(4.55) ∫0T∫ΩK−2♯|v(x,t)|2dxdt≤∫0TZ(t)dt≤c15∫0Tα(t)dtexp(c16∫0Tβ(t)dt)≤cec(A1+B1B3)∫0T∫ωK−1(|vt|2+|∇v|2)dxdt.

Obviously, the right-hand side of (4.55) vanishes as K → ∞. Therefore, passing in (4.55) to the limit, we obtain ∫0T∫Ω|v(x,t)|2dxdt=0, and hence v(x,t)=v1(x,t)−v2(x,t)=0.

Remark 4.1

The solution u of problem (1.1) considered in Theorem 4.2 has the representation u = V + v, where V is a singular part coinciding near the cusp point with the formal asymptotic decomposition of the solution, and v is a regular part having finite energy norm. Theorem 4.2 states only the uniqueness of regular part v. We do not prove the uniqueness of general singular solution of problem (1.1) having a source or sink in the cusp point.

Remark 4.2

The "smallness" assumption of Theorems 4.1 and 4.2 concerns only the smallness of fluxes F(t)=∫∂Ωa(x,t)⋅n(x)dS, i.e. of the magnitude of ∫0T|||F|||J+2dt. We do not suppose that the norms of the boundary value a, initial condition b or the right-hand side f are small.

Acknowledgment

The research of authors was funded by the grant No. S-MIP-17-68 from the Research Council of Lithuania.

Conflict of interest: The authors declare that they have no conflict of interest.

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Received: 2020-06-04

Accepted: 2020-10-17

Published Online: 2021-02-28

This work is licensed under the Creative Commons Attribution 4.0 International License.

Non-stationary Navier–Stokes equations in 2D power cusp domain

II. Existence of the solution

Abstract

1 Introduction

2 Notation, function spaces and auxiliary results

Lemma 2.1

Proof

Lemma 2.2

Proof

Lemma 2.3

Lemma 2.4

Proof

Remark 2.1

3 Stokes problem and Stokes operator

3.1 Estimates of solutions to the Stokes problem

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

3.2 Stokes operator

4 Solvability of Navier–Stokes problem

4.1 Construction of the extension of boundary data

4.2 Existence of the solution in Ω_K

Lemma 4.1

Proof

Theorem 4.1

Proof

4.3 Existence and uniqueness of the solution to problem (4.13)

Theorem 4.2

Proof

Remark 4.1

Remark 4.2

Acknowledgment

References

Journal and Issue

Articles in the same Issue

Non-stationary Navier–Stokes equations in 2D power cusp domain

II. Existence of the solution

Abstract

1 Introduction

2 Notation, function spaces and auxiliary results

Lemma 2.1

Proof

Lemma 2.2

Proof

Lemma 2.3

Lemma 2.4

Proof

Remark 2.1

3 Stokes problem and Stokes operator

3.1 Estimates of solutions to the Stokes problem

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

3.2 Stokes operator

4 Solvability of Navier–Stokes problem

4.1 Construction of the extension of boundary data

4.2 Existence of the solution in ΩK

Lemma 4.1

Proof

Theorem 4.1

Proof

4.3 Existence and uniqueness of the solution to problem (4.13)

Theorem 4.2

Proof

Remark 4.1

Remark 4.2

Acknowledgment

References

Journal and Issue

Articles in the same Issue

4.2 Existence of the solution in Ω_K