We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes

A NONLOCAL STOKES SYSTEM WITH VOLUME CONSTRAINTS

QIANG DU ^∗ AND ZUOQIANG SHI ^†

Abstract. In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition. We prove that the nonlocal model is well-posed and the solution of the nonlocal system converges to the solution of the original Stokes system when the nonlocality vanishes.

1. Introduction. Recently, nonlocal models and corresponding numerical meth- ods attracts lots of attentions and find many successful applications. In solid mechan- ics, the theory of peridynamics [30] has been shown to be an alternative to conven- tional models of elasticity and fracture mechanics. Many numerical methods have also been developed to compute peridynamic model based on solid mathematical analysis [8, 23, 31, 10, 9, 34]. Nonlocal methods are also successfully applied in image process- ing and data analysis [26, 25, 2, 6, 19, 16, 27, 15, 18, 4, 22, 33]. The idea of integral approximation is also applied to derive numerical scheme for solving PDEs on point cloud.

In this paper, we are trying to extend the nonlocal model in Stokes system in fluid mechanics. A nonlocal model was proposed in [11] for Stokes system with periodic boundary condition. In this paper, we consider the no-slip boundary condition. More precisely, for domain Ω⊂Rⁿ,

∆u(x)− ∇p(x) = f(x), x∈Ω

∇ ·u(x) = 0, x∈Ω.

(1.1)

No-slip boundary condition at the boundary is u= 0, at ∂Ω.

(1.2)

For the pressure, we impose average zero condition Z

Ω

p(x)dx= 0.

(1.3)

Comparing to the periodic boundary condition, the no-slip boundary condition is more natural and more often used in real application. However, the theoretical study with no-slip boundary condition is also much more difficult. The first problem is how to enforce no-slip boundary condition in the nonlocal approach. No-slip bound- ary condition is basically Dirichlet type boundary condition. Recently, Du et.al. [8]

proposed volume constraint to deal with the boundary condition in the nonlocal dif- fusion problem. They found that in the nonlocal diffusion problem, since the operator is nonlocal, only enforce the boundary condition on the boundary is not enough, it is necessary to extend the boundary condition to a small region adjacent to the bound- ary. Using this idea, in the nonlocal Stokes system, we extend the no-slip condition to a small layer as shown in Fig. 1. The whole computational domain Ω is decomposed

∗Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA,Email: qd2125@columbia.edu

†Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084. Email:

zqshi@tsinghua.edu.cn.

2δ

V_δ^′ Ω^′δ

Ω = Ω^′δ∪ V_δ^′

∂Ω

Fig. 1.Computational domain in non-local Stokes model.

to two parts Ω =Vδ^′

SΩ^′_δ as shown in Fig. 1 anduis enforced to be zero in V_δ^′, i.e.

(1.4) uδ(x) = 0, x∈ Vδ^′.

In Ω^′_δ, inspired by the point integral method [29], Stokes equation is approximated by an nonlocal approach,







−L^δuδ(x) +G^δpδ(x) = Z

Ω

R¯δ(x,y)f(y)dy, x∈Ω^′_δ, D^δuδ(x)−L¯^δpδ(x) = 0, x∈Ω, (1.5)

The integral operators in (2.1) are defined as L^δu(x) = 1

δ² Z

Ω

Rδ(x,y)(u(x)−u(y))dy, (1.6)

G^δp(x) = 1 2δ²

Z

Ω

Rδ(x,y)(x−y)p(y)dy, (1.7)

D^δu(x) = 1 2δ²

Z

Ω

Rδ(x,y)(x−y)·u(y)dy, (1.8)

L¯^δp(x) = Z

Ω

R¯δ(x,y)(p(x)−p(y))dy.

(1.9) where

Rδ(x,y) =CδR

kx−yk² 4δ²

, R¯δ(x,y) =CδR¯

kx−yk² 4δ²

(1.10)

Cδ is a normalization factor such thatR

RⁿRδ(x,y)dy= 1 and ¯R(r) =R+∞

r R(s)ds.

The kernel function R(r) :R⁺→R⁺ is assumed to beC² smooth and satisfies some mild conditions which are listed in Assumption 1.

Finally, we also need average zero condition for the pressure (1.11)

Z

Ω

pδ(x)dx= 0.

(1.4), (2.1) and (1.11) form a complete nonlocal formulation for Stokes equation.

Nonlocal integral approximation is actually closely related to many numerical schemes of computational fluid dynamics, such as the smoothed particle hydrody- namics (SPH) [14, 20, 21, 24], vortex methods [1, 7] and others [3, 5, 12, 17, 32].

Analysis to the linear steady Stokes equation in this paper could give some new un- derstanding to the theoretical foundation of these methods.

The Stokes system (2.1) is well-known to be a saddle point problem. To preserve the well-posedness, we have to be very careful in the derivation of the nonlocal approx- imation. It is demonstrated that the kernel function has to satisfy some additional conditions for the nonlocal Stokes system with periodic boundary by using Fourier analysis [11]. For the problem we consider in this paper, Fourier transform does not apply. To preserve the well-posedness, we add a relaxation term, ¯L^δpδ(x), in the equation of divergence free. The order of this term isO(δ), so it does not destroy the accuracy of the nonlocal approximation. However, this term is crucial in the analysis of well-posedness.

The rest of the paper is organized as follows. We give the formulating of the nonlocal linear Stokes system in Section 2 and assumptions are also introduced. Then the well-posedness of the nonlocal model is established in Section 3. The vanishing nonlocality limit will be analyzed in Section 4. We will prove the solution of the nonlocal system converges to the solution of the original Stokes system asδ goes to 0. In Section 5, we conclude with a summary and a discussion on future research.

2. Nonlocal Stokes system with volume constraints. Inspired by the vol- ume constraint [8] and the point integral method [29], we consider a nonlocal model of the Stokes system (1.1)-(1.2) as follows:











−L^δuδ(x) +G^δpδ(x) = Z

Ω

R¯δ(x,y)f(y)dy, x∈Ω^′_δ, D^δuδ(x)−L¯^δpδ(x) = 0, x∈Ω,

uδ(x) = 0, x∈ Vδ^′, R

Ωpδ(x)dx = 0.

(2.1)

The integral operators have been defined in (1.6)-(1.9). Here, Ω^′_δ and Vδ^′ are subsets of Ω which are defined as

Ω^′_δ={x∈Ω :B(x,2δ)∩∂Ω =∅}, Vδ^′ = Ω\Ω^′_δ. (2.2)

The relation of Ω,∂Ω, Ω^′_δ andVδ^′ are showed in Fig. 1.

Moreover, we assume that the kernel function R(r) satisfies the conditions in Assumption 1.

Assumption1.

• Assumptions on the computational domain: Ω, ∂Ωare both compact andC^∞ smooth and Ωsatisfies the cone condition.

• Assumptions on the kernel function R(r):

(a) (regularity)R∈C¹(R⁺);

(b) (positivity and compact support)R(r)≥0andR(r) = 0for∀r >1;

(c) (nondegeneracy) ∃δ0>0so that R(r)≥δ0 for0≤r≤ ¹2.

Based on these assumptions, we could prove the well-posedness and the vanishing nonlocality limit of nonlocal Stokes system (2.1).

Remark2.1. If we consider the periodic boundary condition, the well-posedness of the nonlocal system (2.1) is easy to prove.

Applying Fourier transform on (2.1), we have a linear system:

Aδ(ξ)

uˆδ(ξ) ˆ pδ(ξ)

=

¯λδ(ξ) ˆf(ξ) 0

where

Aδ(ξ) =

λδ(ξ)In ibδ(ξ)

−ibδ(ξ)^T cδ(ξ)

i=√

−1,In is ann×n identity matrix and λδ(ξ) = 1

δ² Z

Rⁿ

Rδ(|s|)(1−cos(ξ·s))ds,

¯λδ(ξ) = Z

Rⁿ

R¯δ(|s|) cos(ξ·s)ds, bδ(ξ) = 1

2δ² Z

Rⁿ

sRδ(|s|) sin(ξ·s)ds, cδ(ξ) =

Z

Rⁿ

R¯δ(|s|)(1−cos(ξ·s))ds

Notice that λδ(ξ)>0 andcδ(ξ)>0 using the assumption thatRδ ≥0. Then it is easy to verify that the matrixAδ(ξ) is invertible for anyξ6= 0. The well-posedness follows from the nonsingularity ofAδ(ξ).

3. Well-posedness of the nonlocal Stokes system(2.1). In this section, we will prove the well-posedness of the nonlocal Stokes system (2.1). More precisely, we could prove followng theorem:

Theorem 3.1. Suppose Assumption 1 are satisfied. For anyf ∈H⁻¹(Ω), there exits one and only one pair u, p, such that

(a) u∈H¹(Ω),p∈L²(Ω). In addition,

kuk^H1(Ω)+kpk^L2(Ω)≤Ckfk^H−1(Ω),

whereC >0 is a constant only depends onΩand kernel functionR.

(b) u, p verify the nonlocal Stokes system (2.1).

In the proof of the well-posedness, we need several technical lemmas.

Lemma 3.2. ([29]) Ifδis small enough, for any functionu∈L²(M), there exists a constant C >0 independent onδ andu, such that

Z

Ω

Z

Ω

R

|x−y|² 32δ²

(u(x)−u(y))²dxdy≤C Z

Ω

Z

Ω

R

|x−y|² 4δ²

(u(x)−u(y))²dxdy.

Lemma 3.3. ([28]) For any function u∈L2(Ω^′_δ), there exists a constantC >0 only depneds on Ω, such that

1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)(u(x)−u(y))²dxdy+ 1 δ²

Z

Ω^′_δ

u²(x) Z

Vδ

Rδ(x,y)dy

dx≥C Z

Ω^′_δ|∇v|²dx, where

v(x) = 1 wδ(x)

Z

Ω^′_δ

Rδ(x,y)u(y)dy,

andwδ(x) = Z

Ω

Rδ(x,y) dy.

Proof. First, we introduce an extension ofuto Ω,

˜ u(x) =

u(x), x∈Ω^′_δ, 0, x∈ Vδ^′. It follows that

v(x) = 1 wδ(x)

Z

Ω^′_δ

Rδ(x,y)u(y)dy= 1 wδ(x)

Z

Ω

Rδ(x,y)˜u(y)dy Notice that for any x ∈ Ω^′_δ, wδ(x) = 1 and

Z

Ω∇^xRδ(x,y)dy = 0. Then for any x∈Ω^′_δ, we have

∇v(x) = 1 wδ(x)

Z

Ω

(∇^xRδ(x,y))˜u(y)dy=− 1 wδ(x)

Z

Ω

(∇^xRδ(x,y))(˜u(x)−u(y))dy˜ This gives that

Z

Ω^′_δ|∇v(x)|²dx= 1

¯ w_δ²

Z

Ω^′_δ

Z

Ω

(∇^xRδ(x,y))(˜u(x)−u(y))dy˜

2

dx

≤ 1

¯ w_δ²

Z

Ω^′_δ

Z

Ω|∇^xRδ(x,y)|dy Z

Ω|∇^xRδ(x,y)|(˜u(x)−u(y))˜ ²dy

dx

≤ C δ²

Z

Ω^′_δ

Z

Ω

Rδ(x,y)(˜u(x)−u(y))˜ ²dy

dx

= C δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)(u(x)−u(y))²dxdy+ C δ²

Z

V_δ^′

Z

Ω^′_δ

Rδ(x,y)u(y)²dy

! dx which complete the proof.

Lemma 3.4. ([28]) For any function u∈L2(Ω^′_δ), there exists a constantC >0 independent onδ, such that

1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)(u(x)−u(y))²dxdy+ Z

Ω^′_δ

u²(x) Z

V_δ^′

Rδ(x,y)dy

!

dx≥Ckuk²L2(Ω^′_δ), as long asδ small enough.

Now we can prove the main theorem in this section, Theorem 3.1.

Proof. [Proof of Theorem 3.1:]

First, in the nonlocal Stokes system, we replace the conditionR

Ωpδ(x)dx= 0 by R

Ω^′_δpδ(x)dx= 0 and denote the pressure in the original nonlocal Stokes system as ¯p.

It is obvious that

¯

pδ=pδ− 1

|Ω| Z

Ω

pδ(x)dx (3.1)

Multiplying uδ on the first equation of (2.1) and multiplying p on the third equation of (2.1) and adding them together, we can get

1 δ²

Z

Ω

Z

Ω

Rδ(x,y)|uδ(x)−uδ(y)|²dxdy+

Z

Ω

Z

Ω

R¯δ(x,y)(pδ(x)−pδ(y))²dxdy

=−2 Z

Ω

Z

Ω

R¯δ(x,y)f(y)·uδ(x)dxdy (3.2)

Using this equation, we can get the uniqueness of the solution of the nonlocal Stokes equations (2.1) which also imply the existence of the solution in L² using standard theory for integral equations.

From (3.2), using Lemma 3.4, there existsC >0, such that kuδk²L²(Ω)≤Ckfk^H−1(Ω)kuδk^H1(Ω). (3.3)

In addition, from the first equation of (2.1), uδ has following expression, for any x∈Ω^′_δ,

uδ(x) = 1 wδ(x)

Z

Ω

Rδ(x,y)uδ(y)dy+ 1 2wδ(x)

Z

Ω

Rδ(x,y)(x−y)pδ(y)dy− δ² wδ(x)

Z

Ω

R¯δ(x,y)f(y)dy (3.4)

Using Lemma 3.3 and (3.2), (3.3), we have k∇

1 wδ(x)

Z

Ω

Rδ(x,y)uδ(y)dy

k²L²(Ω^′_δ)

≤C δ²

Z

Ω

Z

Ω

Rδ(x,y)|uδ(x)−uδ(y)|²dxdy≤Ckfk^H−1(Ω)kuδk^H1(Ω) (3.5)

Notice that for anyx∈Ω^′_δ,wδ(x) is a positive constant. Then we have

k∇

1 2wδ(x)

Z

Ω

Rδ(x,y)(x−y)pδ(y)dy

k²L²(Ω^′_δ)

(3.6)

≤C Z

Ω^′_δ

Z

Ω∇^xRδ(x,y)(x−y)pδ(y)dy

2

dx+C Z

Ω^′_δ

Z

Ω

Rδ(x,y)pδ(y)dy 2

dx

≤C δ²

Z

Ω

Z

Ω|R^′_δ(x,y)||x−y|²pδ(y)dy

2

dx+C Z

Ω

Z

Ω

Rδ(x,y)pδ(y)dy 2

dx

≤C Z

Ω

Z

Ω|R^′_δ(x,y)|pδ(y)dy 2

dx+C Z

Ω

Z

Ω

Rδ(x,y)pδ(y)dy 2

dx

≤Ckpk²L²(Ω). where

R^′_δ(x,y) =CδR^′

|x−y|² 4δ²

, R^′(r) = d drR(r).

In addition, direct calculation gives that k∇

δ² wδ(x)

Z

Ω

R¯δ(x,y)f(y)dy

k^L2(Ω′δ)≤Ckfk^H−1(Ω) (3.7)

Putting (3.3)-(3.7) together, we obtain

kuδkH¹(Ω)≤CkfkH⁻¹(Ω)+CkpδkL²(Ω)

(3.8)

Next, we turn to estimate the pressurep. First, considering the problem

∇ ·v(x) =pδ(x), x∈Ω^′_δ. (3.9)

It is well known (e.g. Section 3.3 of [13]) that if Ω^′_δ satisfies cone condition, there exists at least one solution of (3.9), denoted byv, such that

v∈H₀¹(Ω^′_δ), kvkH¹(Ω^′_δ)≤ckpkL²(Ω^′_δ)

(3.10)

Then, we extend v to Ω by assigning the value on Vδ^′ to be 0 and denote the new function also byv. Obviously, we have

v ∈H₀¹(Ω^′_δ)∩H₀¹(Ω), kvkH¹(Ω)≤ckpδkL²(Ω^′_δ)

(3.11)

On the other hand, using the second equation of (2.1),∀x∈Ω

¯

wδ(x)pδ(x) = Z

Ω

R¯δ(x,y)pδ(y)dy+ 1 2δ²

Z

Ω

Rδ(x,y)(x−y)·uδ(y)dy

= Z

Ω^′_δ

R¯δ(x,y)∇ ·v(y)dy+ 1 2δ²

Z

Ω^′_δ

Rδ(x,y)(x−y)·uδ(y)dy + 1

2δ² Z

V_δ^′

Rδ(x,y)(x−y)·uδ(y)dy+ Z

V_δ^′

R¯δ(x,y)pδ(y)dy

=− 1 2δ²

Z

Ω^′_δ

Rδ(x,y)(x−y)·v(y)dy¯ + Z

V_δ^′

R¯δ(x,y)pδ(y)dy (3.12)

where ¯wδ(x) =R

ΩR¯δ(x,y)dy and ¯v=v−uδ. Then, it follows that

1 2δ²

Z

Ω^′_δ

v(x)¯ Z

Ω

Rδ(x,y)(x−y)pδ(y)dy

dx

=− 1 2δ²

Z

Ω

pδ(x) Z

Ω^′_δ

Rδ(x,y)(x−y)¯v(y)dy

! dx

= Z

Ω

p²_δ(x) ¯wδ(x)dx− Z

Ω

pδ(x) Z

V_δ^′

R¯δ(x,y)pδ(y)dy

! dx.

(3.13)

The first term is positive which is a good term. The second term becomes

− Z

Ω

pδ(x) Z

V_δ^′

R¯δ(x,y)pδ(y)dy

! dx (3.14)

= Z

Ω

pδ(x) Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))dy

! dx−

Z

Ω

p²_δ(x) Z

V_δ^′

R¯δ(x,y)dy

! dx.

The second term of (3.14) can be controlled by the first term of (3.13). And the first

term is bounded by

Z

Ω

pδ(x) Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))dy

! dx (3.15)

=1 2

Z

V_δ^′

Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))²dydx+ Z

Ω^′_δ

pδ(x) Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))dy

! dx

≥1 2

Z

V_δ^′

Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))²dydx+ Z

Ω^′_δ

Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))²dy

! dx

− Z

Ω^′_δ

Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))pδ(y)dy

! dx

≥1 2

Z

Ω

Z

V_δ^′

R¯δ(x,y)(pδ(x)−pδ(y))²dy

! dx−1

2 Z

V_δ^′

p²_δ(x) Z

Ω^′_δ

R¯δ(x,y)dy

! dx.

Combining (3.13)-(3.15), we get

1 2δ²

Z

Ω^′_δ

¯ v(x)

Z

Ω

Rδ(x,y)(x−y)pδ(y)dy

dx≥ Z

Ω^′_δ

p²_δ(x) Z

Ω^′_δ

R¯δ(x,y)dy

! dx (3.16)

Now, we are ready to get the estimate ofpδ. Multiplying ¯v on both sides of the first equation of (2.1) and integrating over Ω^′_δ, using the fact that ¯v(x) = 0, x∈ Vδ^′, we have

− 1 2δ²

Z

Ω

Z

Ω

Rδ(x,y)(uδ(x)−uδ(y))·(¯v(x)−v(y))dxdy+¯ 1 2δ²

Z

Ω^′_δ

¯ v(x)

Z

Ω

Rδ(x,y)(x−y)pδ(y)dy

dx

= Z

Ω^′_δ

¯ v(x)

Z

Ω

R¯δ(x,y)f(y)dy

dx (3.17)

Using (3.2), (3.8), (3.11), (3.17) and (3.16), we have 1

2kpk²L²(Ω^′_δ)

≤ 1

2δ² Z

Ω

Z

Ω

Rδ(x,y)|uδ(x)−uδ(y)|²dxdy 1/2

1 2δ²

Z

Ω

Z

Ω

Rδ(x,y)|v(x)¯ −v(y)¯ |²dxdy 1/2

+kv¯kH¹(Ω^′_δ)kfkH⁻¹(Ω)

≤ 1

2δ² Z

Ω

Z

Ω

Rδ(x,y)|uδ(x)−uδ(y)|²dxdy

1/2 1 2δ²

Z

Ω

Z

Ω

Rδ(x,y)|v(x)−v(y)|²dxdy 1/2

+ 1

2δ² Z

Ω

Z

Ω

Rδ(x,y)|uδ(x)−uδ(y)|²dxdy 1/2!

+ (kvkH¹(Ω^′_δ)+kuδkH¹(Ω^′_δ))kfkH⁻¹(Ω)

≤kuδk^H1(Ω)kfk^H−1(Ω)+kuδk^1/2H¹(Ω)kfk^1/2H⁻¹(Ω)kvk^H1(Ω)+C(kpδk^L2(Ω′δ)+kfk^H−1(Ω))kfk^H−1(Ω)

≤C(kpδk^L2(Ω)+kfkH⁻¹(Ω))kfkH⁻¹(Ω). (3.18)

On the other hand, using Lemma 3.2, kpk^L2(Ω) can be bounded by kpk^L2(Ω′δ). First, notice that using the nondegeneracy assumption in Assumption 1, it is easy to verify that for anyx∈Ω,

(3.19)

Z

Ω^′_δ

R¯4δ(x,y)dy≥c0>0.

where

R¯4δ(x,y) =CδR¯

kx−yk² 4(4δ)²

, Cδ is the normalization factor in (1.10).

kpδk²L²(Ω)≤C Z

Ω

Z

Ω^′_δ|pδ(x)|²R¯4δ(x,y)dy

! dx

≤C Z

Ω

Z

Ω^′_δ|pδ(x)−pδ(y)|²R¯4δ(x,y)dy

! dx+C

Z

Ω

Z

Ω^′_δ|pδ(y)|²R¯4δ(x,y)dy

! dx

≤C Z

Ω

Z

Ω|pδ(x)−pδ(y)|²R¯4δ(x,y)dydx+C Z

Ω^′_δ|pδ(y)|²dx

≤C Z

Ω

Z

Ω|pδ(x)−pδ(y)|²R¯δ(x,y)dydx+Ckpδk²L²(Ω^′_δ)

Using (3.2), (3.8), (3.18) and (3.20), we have

kpδk²L²(Ω)≤Ckuδk^H1(Ω)kfk^H−1(Ω)+Ckpδk²L²(Ω^′_δ)

≤C(kpδk^L2(Ω)+kfk^H−1(Ω))kfk^H−1(Ω) (3.20)

Together with (3.18), we have

kpδkL²(Ω)≤CkfkH⁻¹(Ω). (3.21)

This also gives theH¹ estimate ofuδ using (3.8), kuδk^H1(Ω)≤Ckfk^H−1(Ω). (3.22)

and using (3.1)

kp¯δk^L2(Ω)≤ kpδk^L2(Ω)+ 1

|Ω| Z

Ω

pδ(x)dx

≤Ckfk^H−1(Ω), (3.23)

here we use the fact that 1

|Ω| Z

Ω

pδ(x)dx ≤ 1

p|Ω|kpδk^L2(Ω).

4. Vanishing nonlocality. Besides the well-posedness, another problem we are interested is the limit behavior of the nonlocal Stokes system (2.1) as the nonlocality vanish, i.e. δ→0. In this section, we will answer this question. Under some assump- tions, we can prove that the solution of the nonlocal Stokes system converges to the solution of the Stokes system as δ→ 0. Furthermore, we could give an estimate of the convergence rate. The result is summarized in Theorem 4.2.

Before stating the main theorem, we give several lemmas which will be used in proving the main theorem.

We also need following theorem regarding the nonlocal approximation of the Laplace operator. And the proof can be found in [28].

Theorem 4.1. ([28]) Let u∈H³(Ω) and r(x) =−1

δ² Z

Ω

Rδ(x,y)(u(x)−u(y))dy− Z

Ω

R¯δ(x,y)∆u(y)dy, ∀x∈Ω^′_δ.

There exists constants C, T0 depending only onΩ, so that for anyδ≤T0, kr(x)kL²(Ω^′_δ)≤CδkukH³(Ω),

(4.1)

k∇r(x)kL²(Ω^′_δ)≤Ckuk^H3(Ω). (4.2)

This is the main result in this section regarding the convergence of the nonlocal Stokes system as the nonlocality vanish.

Theorem 4.2. Letu(x),p(x)be solution of Stokes system(1.1)anduδ(x), pδ(x) be solution of nonlocal Stokes system (2.1)withf ∈H¹(Ω). There exists C >0only depends onΩandR, such that

ku−uδk^H1(Ω′δ)+kp−pδk^L2(Ω)≤C√

δkfk^H1(Ω)

Proof. Let eδ(x) =u(x)−uδ(x) and dδ =p−pδ− |Ω^1′_δ|

R

Ω^′_δ(p(x)−pδ(x))dx, theneδ, dδ satisfies



















−1 δ²

Z

Ω

Rδ(x,y)(eδ(x)−eδ(y))dy+ 1 2δ²

Z

Ω

Rδ(x,y)(x−y)dδ(y)dy = ru(x), x∈Ω^′_δ, eδ(x) = u(x), x∈ Vδ^′, 1

2δ² Z

Ω

Rδ(x,y)(x−y)·eδ(y)dy− Z

Ω

R¯δ(x,y)(dδ(x)−dδ(y))dy = rp(x), x∈Ω, Z

Ω^′_δ

dδ(x)dx = 0, (4.3)

where

r_u(x) = Z

Ω

R¯δ(x,y)∆u(y)dy+ 1 δ²

Z

Ω

Rδ(x,y)(u(x)−u(y))dy, ∀x∈Ω^′_δ (4.4)

rp(x) =− Z

Ω

R¯δ(x,y)(p(x)−p(y))dy, ∀x∈Ω.

(4.5)

First, we focus on the following estimate

1 δ²

Z

Ω^′_δ

eδ(x)· Z

Ω

Rδ(x,y)(eδ(x)−eδ(y))dydx (4.6)

=1 δ²

Z

Ω^′_δ

eδ(x)· Z

Ω^′_δ

Rδ(x,y)(eδ(x)−eδ(y))dydx+ 1 δ²

Z

Ω^′_δ

eδ(x)· Z

V_δ^′

Rδ(x,y)(eδ(x)−eδ(y))dydx

= 1 2δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy+ 1 δ²

Z

Ω^′_δ

eδ(x)· Z

V_δ^′

Rδ(x,y)(eδ(x)−eδ(y))dydx.

The second term of the right hand side of (4.6) can be calculated as

1 δ²

Z

Ω^′_δ

eδ(x)· Z

V_δ^′

Rδ(x,y)(eδ(x)−eδ(y))dydx (4.7)

=1 δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

! dx− 1

δ² Z

Ω^′_δ

eδ(x)· Z

V_δ^′

Rδ(x,y)u(y)dy

! dx.

Here we use the definition ofeδ and the volume constraint conditionuδ(x) = 0, x∈ Vδ^′ to get thateδ(x) =u(x), x∈ Vδ^′.

The first term is positive which is good for us. We only need to bound the second term of (4.7) to show that it can be controlled by the first term. First, the second term can be bounded as following

1 δ²

Z

Ω^′_δ

eδ(x)· Z

V_δ^′

Rδ(x,y)u(y)dy

! dx

(4.8)

≤1 δ²

Z

Ω^′_δ|eδ(x)| Z

V_δ^′

Rδ(x,y)dy

!1/2

Z

V_δ^′

Rδ(x,y)|u(y)|²dy

!1/2

dx

≤1 δ²

Z

Ω^′_δ

1

2|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

! dx+ 2

Z

Ω^′_δ

Z

V_δ^′

Rδ(x,y)|u(y)|²dy

! dx

!

≤ 1 2δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

!

dx+ 2 δ²

Z

V^′_δ|u(y)|² Z

Ω^′_δ

Rδ(x,y)dx

! dy

≤ 1 2δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

!

dx+ C δ²

Z

V^′_δ|u(y)|²dy

≤ 1 2δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

!

dx+Cδkfk²H¹(Ω).

Here we use Lemma A.1 in Appendix A to get the last inequality.

By substituting (4.8), (4.7) in (4.6), we get

1 δ²

Z

Ω^′_δ

eδ(x)· Z

Ω

Rδ(x,y)(eδ(x)−eδ(y))dydx (4.9)

≥ 1 2δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy

+ 1 2δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

!

dx−Ckfk²H¹(Ω)δ.

This is the key estimate we used to get convergence.

We also need following bound

1 δ²

Z

Ω^′_δ

eδ(x)· Z

Ω

Rδ(x,y)(x−y)dδ(y)dy

dx+ 1 δ²

Z

Ω

dδ(x) Z

Ω

Rδ(x,y)(x−y)·eδ(y)dy

dx (4.10)

= 1 δ²

Z

Ω

dδ(x) Z

V_δ^′

Rδ(x,y)(x−y)·eδ(y)dy

! dx

≤1 δ

Z

Ω

Z

V_δ^′

Rδ(x,y)|dδ(x)||u(y)|dy

! dx

≤1 δ

"

Z

Ω

Z

V^′_δ

Rδ(x,y)|dδ(x)|²dy

! dx

Z

Ω

Z

V_δ^′

Rδ(x,y)|u(y)|²dy

! dx

#1/2

≤C√

δkfk^H1(Ω)kdδk^L2(Ω).

Multiplying eδ(x), dδ(x) on both sides of the first and third equation of (4.3) and integrating over Ω^′_δ, Ω respectively and adding them together, using (4.9), (4.10), we have

1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy+ 1 2δ²

Z

Ω^′_δ|eδ(x)|² Z

V_δ^′

Rδ(x,y)dy

! dx (4.11)

+ Z

Ω

Z

Ω

Rδ(x,y)|dδ(x)−dδ(y)|²dxdy

≤kruk^L2(Ω′δ)keδk^L2(Ω′δ)+krpk^L2(Ω)kdδk^L2(Ω)+C√

δkfk^H1(Ω)kdδk^L2(Ω)+Cδkfk²H¹(Ω). To simplify the notation, we denote the right hand side of (4.11) asQ².

It is well known (e.g. Section 3.3 of [13]) that with the condition thatR

Ω^′_δdδ(x)dx= 0, there exists at least one functionψ, such that

∇ ·ψ(x) =dδ(x), x∈Ω^′_δ, and ψ∈H₀¹(Ω^′_δ), kψkH¹(Ω^′_δ)≤ckdδkL²(Ω^′_δ)

(4.12)

Then, we extend ψ to Ω by assigning the value on Vδ^′ to be 0 and denote the new function also byψ. Obviously, we have

ψ∈H₀¹(Ω^′_δ)∩H₀¹(Ω), kψk^H1(Ω)≤ckdδk^L2(Ω′δ)

(4.13)

Using the third equation of (4.3), we have

¯

wδ(x)dδ(x) = Z

Ω

R¯δ(x,y)dδ(y)dy+ 1 2δ²

Z

Ω

Rδ(x,y)(x−y)·eδ(y)dy−rp(x)

= Z

Ω^′_δ

R¯δ(x,y)∇ ·ψ(y)dy+ 1 2δ²

Z

Ω^′_δ

Rδ(x,y)(x−y)·eδ(y)dy + 1

2δ² Z

V_δ^′

Rδ(x,y)(x−y)·eδ(y)dy+ Z

V_δ^′

R¯δ(x,y)dδ(y)dy−rp(x)

=− 1 2δ²

Z

Ω^′_δ

Rδ(x,y)(x−y)·ψ(y)dy¯ + 1 2δ²

Z

V_δ^′

Rδ(x,y)(x−y)·u(y)dy +

Z

V_δ^′

R¯δ(x,y)dδ(y)dy−rp(x) (4.14)

where ¯wδ(x) =R

ΩR¯δ(x,y)dy and ¯ψ=ψ−eδ. Then, it follows that

1 2δ²

Z

Ω^′_δ

ψ(x)¯ Z

Ω

Rδ(x,y)(x−y)dδ(y)dy

dx

=− 1 2δ²

Z

Ω

dδ(x) Z

Ω^′_δ

Rδ(x,y)(x−y) ¯ψ(y)dy

! dx

= Z

Ω

d²_δ(x) ¯wδ(x)dx− Z

Ω

dδ(x) Z

V_δ^′

R¯δ(x,y)dδ(y)dy

! dx

− Z

Ω

dδ(x) Z

V_δ^′

Rδ(x,y)(x−y)·u(y)dy

! dx+

Z

Ω

dδ(x)rp(x)dx (4.15)

The first term is positive which is a good term. The second term becomes

− Z

Ω

dδ(x) Z

V_δ^′

R¯δ(x,y)dδ(y)dy

! dx (4.16)

= Z

Ω

dδ(x) Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))dy

!

dx− 1 2δ²

Z

Ω

d²_δ(x) Z

V_δ^′

R¯δ(x,y)dy

! dx.

The second term of (4.16) can be controlled by the first term of (4.15). And the first

term is bounded by

Z

Ω

dδ(x) Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))dy

! dx (4.17)

=1 2

Z

V_δ^′

Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))²dydx+ Z

Ω^′_δ

dδ(x) Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))dy

! dx

≥1 2

Z

V_δ^′

Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))²dydx+ Z

Ω^′_δ

Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))²dy

! dx

− Z

Ω^′_δ

Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))dδ(y)dy

! dx

≥1 2

Z

Ω

Z

V_δ^′

R¯δ(x,y)(dδ(x)−dδ(y))²dy

! dx−1

2 Z

V_δ^′

d²_δ(x) Z

Ω^′_δ

R¯δ(x,y)dy

! dx.

Combining (4.15)-(4.17), we get

1 2δ²

Z

Ω^′_δ

ψ(x)¯ Z

Ω

Rδ(x,y)(x−y)dδ(y)dy

dx (4.18)

≥ Z

Ω^′_δ

d²_δ(x) Z

Ω^′_δ

R¯δ(x,y)dy

!

dx− 1 2δ²

Z

Ω

dδ(x) Z

V^′_δ

Rδ(x,y)(x−y)·u(y)dy

! dx +

Z

Ω

dδ(x)rp(x)dx.

In addition, we have

1 2δ²

Z

Ω

dδ(x) Z

V_δ^′

Rδ(x,y)(x−y)·u(y)dy

! dx

(4.19)

≤1 2δ

Z

Ω|dδ(x)| Z

V_δ^′

Rδ(x,y)|u(y)|dy

! dx

≤1 2δ

"

Z

Ω|dδ(x)|² Z

V_δ^′

Rδ(x,y)dy

! dx

Z

Ω

Z

V_δ^′

Rδ(x,y)|u(y)|²dy

! dx

#1/2

≤C√

δkfkH¹(Ω)kdδkL²(Ω). and

Z

Ω

dδ(x)rp(x)dx (4.20)

= Z

Ω

dδ(x) Z

Ω

R¯δ(x,y)(p(x)−p(y))dy

dx

≤CδkpkH¹(Ω)kdδkL²(Ω)

≤Cδkfk^H1(Ω)kdδk^L2(Ω).

Multiplying ¯ψon both sides of the first equation of (4.3) and using (4.18), (4.19), (4.20), we have

kdδk²L²(Ω^′_δ)≤1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)(eδ(x)−eδ(y))·( ¯ψ(x)−ψ(y))dxdy¯

+ 1 δ²

Z

Ω^′_δ

ψ(x)¯ · Z

V^′_δ

Rδ(x,y)(eδ(x)−eδ(y))dy

! dx +kψ¯k^L2(Ω′δ)kruk^L2(Ω)+C√

δkfk^H1(Ω)kdδk^L2(Ω) (4.21)

The first term can be bounded as

1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)(eδ(x)−eδ(y))·( ¯ψ(x)−ψ(y))dxdy¯ (4.22)

≤ 1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy

!1/2

1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|ψ(x)¯ −ψ(y)¯ |²dxdy

!1/2

≤ 1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy

!1/2

 1 δ²

Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|ψ(x)−ψ(y)|²dxdy

!1/2

+ 1

δ² Z

Ω^′_δ

Z

Ω^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dxdy

!1/2



≤Q²+CQkψk^H1(Ω′δ)≤Q²+CQkdδk^L2(Ω′δ).

The estimate of the second term of (4.21) is a little involved. First

1 δ²

Z

Ω^′_δ

ψ(x)¯ · Z

V_δ^′

Rδ(x,y)(eδ(x)−eδ(y))dy

! dx

(4.23)

≤ 1 δ²

Z

Ω^′_δ

Z

V_δ^′

Rδ(x,y)( ¯ψ(x)−ψ(y))¯ ·(eδ(x)−eδ(y))dy

! dx

+

1 δ²

Z

V_δ^′

u(x)· Z

Ω^′_δ

Rδ(x,y)(eδ(x)−eδ(y))dy

! dx

≤



 1 δ²

Z

Ω^′_δ

Z

V^′_δ

Rδ(x,y)|ψ(x)¯ −ψ(y)¯ |²dy

! dx

!1/2

+ 1

δ² Z

V_δ^′|u(x)|² Z

Ω^′_δ

Rδ(x,y)dy

! dx

!1/2

 1

δ² Z

Ω^′_δ

Z

V^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dy

! dx

!1/2

≤C

kψk^H1(Ω)+√

δkfk^H1(Ω) 1 δ²

Z

Ω^′_δ

Z

V^′_δ

Rδ(x,y)|eδ(x)−eδ(y)|²dy

! dx

!1/2

+ C δ²

Z

Ω^′_δ

Z

V_δ^′

Rδ(x,y)|eδ(x)−eδ(y)|²dy

! dx.