Boundary conditions for the Stokes problem and a pressure-Poisson problem
Kazunori Matsui


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ℝ
ℕ
H^1(Ω)
H^1_0(Ω)
H^1_Γ_i(Ω)
H^1_Γ_1(Ω)
H^1_Γ_2(Ω)
L^2(Ω)
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div
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∫_Γ
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spmpsci





Division of Mathematical and Physical Sciences,

Graduate School of Natural Science and Technology,
 
Kanazawa University, Kakuma, Kanazawa 920-1192 Japan


Boundary conditions for the Stokes problem and
 a pressure-Poisson problem
    Kazunori Matsui
    December 30, 2023
==========================================================================


  Email address: first-lucky@stu.kanazawa-u.ac.jp

    Date: December 30, 2023



  We consider a boundary value problem for the stationary Stokes problem
  and the corresponding pressure-Poisson equation.
  We propose a new formulation for the pressure-Poisson problem 
  with an appropriate additional boundary condition.
  We establish error estimates between solutions to the Stokes problem 
  and the pressure-Poisson problem in terms of the additional boundary condition.
  As boundary conditions for the Stokes problem, 
  we use a traction boundary condition and a pressure boundary condition 
  introduced in C. Conca et al (1994).



Key words: Stokes problem, Pressure-Poisson equation

2010 MSC: 35Q35, 35A35, 76D07






§ INTRODUCTION


Let Ω be a bounded connected open set of ^3
with Lipschitz continuous boundary Γ.
We assume that there exist two relatively open subsets
Γ_1 and Γ_2 of Γ such that

    |Γ∖(Γ_1∪Γ_2)|=0,  
      |Γ_1|,|Γ_2|> 0,  Γ_1∩Γ_2=∅,  Γ_1=Γ_1,  Γ_2=Γ_2,

where A is the closure of A⊂Γ with respect to Γ,
A is the interior of A with respect to Γ
and |A| is the two dimensional Hausdorff measure.

The strong form of the Stokes problem is given as follows. 
Find u^S:Ω→^3 and p^S:Ω→ such that

    S{[ -Δ u^S+∇ p^S            =            F           Ω,;          u^S            =            0           Ω,;          u^S            =            0         Γ_1,; T_ν(u^S,p^S)            =          t^b         Γ_2, ].

holds, where t^b:Γ_2→^3,
ν is the unit outward normal vector for Γ and

    T_ν(u,p)_i:=
      ∑^3_j=1(∂ u_i/∂ x_j+∂ u_j/∂ x_i)ν_j-pν_i
        i=1,2,3.

The functions u^S and p^S are the velocity and the pressure of the flow
governed by (<ref>), respectively.
We refer to <cit.> and <cit.> for the details on the Stokes problem
(i.e., physical background and corresponding mathematical analysis).
Taking the divergence of the first equation, we obtain

    F=(-Δ u^S+∇ p^S)=-Δ ( u^S)+Δ p^S=Δ p^S.

This equation is often called the pressure-Poisson equation and 
is used in numerical schemes such as MAC (marker and sell), 
SMAC (simplified MAC) or the projection method 
(see, e.g., <cit.>).

We need an additional boundary condition for solving the equation (<ref>). 
In the real-would applications, the additional boundary condition
is usually given by using experimental or plausible values.
We consider the following boundary value problem for 
the pressure-Poisson equation:
Find u^PP:Ω→^3 and p^PP:Ω→ satisfying

    PP{[ -Δ u^PP-∇( u^PP)+∇ p^PP                       =                       F                      Ω,;                 -Δ p^PP                       =                     - F                      Ω,;                    u^PP                       =                       0                    Γ_1,;               ∂ p^PP/∂ν                       =                     g^b                    Γ_1,;          T_ν(u^PP,p^PP)                       =                     t^b                    Γ_2,;                    p^PP                       =                     p^b                    Γ_2, ].

where g^b:Γ_1→, p^b:Γ_2→
are the data for the additional boundary conditions.
We call this problem the pressure-Poisson problem.
The second term -∇( u^PP) in the first equation of (<ref>) 
is necessary to treat the traction boundary condition 
in a weak formulation.
The idea of using (<ref>) instead of u^S=0 is useful
for calculating the pressure numerically in the Navier–Stokes problem.
For example, this idea is used in the MAC, SMAC and projection methods.

In this paper, we establish error estimates between solutions 
for (<ref>) and (<ref>) in terms of 
the additional boundary conditions.
As the boundary condition for the Stokes problem, 
we also consider the boundary condition introduced in <cit.>;

    {[    u    =    0 Γ_1,;  u×ν    =    0 Γ_2,;    p    =  p^b Γ_2, ].

where “×” is the cross product in ^3
(see also <cit.>).
Since boundary conditions which contain a Dirichlet boundary condition
for the pressure often appear in engineering problems,
a comparison between (<ref>) and the Stokes problem with (<ref>) 
is important.
For example, an end of pipe such as blood vessels or pipelines 
corresponds to the boundary Γ_2 (Fig. <ref>).


The organization of this paper is as follows.
In Section <ref> we introduce notations and symbols used in this work
and the weak form of these problems. We also prove the well-posedness of 
the problems (<ref>) and (<ref>) and show several properties of them.
In Section <ref> we establish error estimates between solutions to 
the problems (<ref>) and (<ref>) in terms of 
the additional boundary conditions.
Section <ref> is devoted to the study of the Stokes problem with 
the boundary condition (<ref>).
We conclude this paper with several comments on future works 
in Section <ref>.





§ PRELIMINARIES



 §.§ Notation




We use the usual Lebesgue space  and Sobolev spaces 
H^r(Ω) =W^r,2(Ω) for a non-negative integer r,
together with their standard norms.
For spaces of vector-valued functions, we write ^3, and so on.
The space  denotes the closure of C^∞_0(Ω) in .
𝒟'(Ω) denotes the space of distributions on Ω.
We set

    [                                                        :=      {ψ∈ | ψ=0Γ_i}  (i=1,2),;                            H                           := {φ∈^3 | φ=0Γ_1,
      φ×ν=0Γ_2}. ]


We also use the Lebesgue space L^2(Γ) and Sobolev space H^1/2(Γ)
defined on Γ.
The norm η_H^1/2(Γ) is defined by

    η_H^1/2(Γ)
        :=(η^2_L^2(Γ)+∫_Γ∫_Γ|η(x)-η(y)|^2/|x-y|^3ds(x)ds(y))^1/2,

where ds denotes the surface measure of Γ.
For function spaces defined on Γ_i (i=1,2), we write L^2(Γ_i), and so on.

We further set 

    [ p_,i:=∂ p/∂ x_i,  
      p_,ij:=∂ p/∂ x_i∂ x_j,  
      D_ij(u):=1/2(u_i,j+u_j,i),   D(u):D(φ):=∑^3_l,m=1D_lm(u)D_lm(φ) ]

for all p:Ω→,u=(u_1,u_2,u_3):Ω→^3,
φ=(φ_1,φ_2,φ_3):Ω→^3 and i,j=1,2,3.





 §.§ Preliminary results




Let γ_0∈ℒ(, H^1/2(Γ)) be
the standard trace operator.
The trace operator γ_0 is surjective and satisfies 
(γ_0)= <cit.>.
Let ν be the unit outward normal for Γ.
Since ν is a unit vector, 
^3∋ u↦ u·ν:=(γ_0 u)·ν∈ H^1/2(Γ)
is a linear continuous map.
For all u∈^3 and p∈, the following Gauss divergence formula holds:

    u·∇ p+( u)p=∫_Γ(u·ν)p.


For i=1,2, composition of the trace operator γ_0 and 
the restriction H^1/2(Γ)→ H^1/2(Γ_i) 
is denoted by ψ↦ψ|_Γ_i. 
This map is continuous from  to H^1/2(Γ_i).
Since the kernel of this map is , there exists a constant c>0 such that

    ψ_/≤ cψ|_Γ_i_H^1/2(Γ_i),

where ψ_/:=inf_q∈ψ+q_.
We simply write ψ instead of ψ|_Γ_i
when there is no ambiguity.
We denote by ⟨·,·⟩_Γ_i the duality pairing between
H^-1/2(Γ_i):=H^1/2(Γ_i)^* and H^1/2(Γ_i).
We remark that η^*∈ L^2(Γ_i) can be identified with 
a element of H^-1/2(Γ_i) by

    ⟨η^*,η⟩_Γ_i:=∫_Γ_iη^*η    η∈ H^1/2(Γ_i).


For u∈^3 and p∈ satisfying Δ u+∇( u)∈
and Δ p∈, we set 

    [                 ⟨ D(u)ν,φ⟩_Γ_2                             := (D(u):D(φ)
      +1/2(Δ u+∇( u))φ)                          φ∈^3,;                 ⟨∂ p/∂ν,ψ⟩_Γ_1                             :=                (∇ p·∇ψ+(Δ p)ψ)                            ψ∈. ]

We remark that u∈ H^2(Ω)^3 and p∈ H^2(Ω) satisfy

    ⟨ D(u)ν,φ⟩_Γ_2
      =∫_Γ_2(∑^3_i,j=1D_ij(u)φ_i ν_j),  ⟨∂ p/∂ν,ψ⟩_Γ_1
      =∫_Γ_1∂ p/∂νψ

for all φ∈^3 and ψ∈.
For u∈^3 and p∈ satisfying Δ u+∇( u)∈, we set 

    ⟨ T_ν(u,p)ν,φ⟩_Γ_2
      :=2⟨ D(u)ν,φ⟩_Γ_2-∫_Γ_2pφ·ν    φ∈^3.


We recall the following five theorems which are necessary for
the existence and the uniqueness of a solution to the Stokes problem.


  <cit.>
Let (X,·_X) and (Q,·_Q) be two real Hilbert spaces.
Let a:X× X→ and b:X× Q→ be
bilinear and continuous maps and let f∈ X^*.
If there exist two constants α>0 and β>0 such that

    [                 a(v,v)                      ≥                 αv^2_X                  v∈ V,; sup_v∈ X∖{0}b(v,q)/v_X                      ≥                   βq_Q                  q∈ Q, ]

where V={v∈ X | b(v,q)=0q∈ Q},
then there exists a unique solution (u,p)∈ X× Q to the following problem:

    {[ a(u,v)+b(v,p)             =          f(v)         v∈ X,;        b(u,q)             =             0         q∈ Q. ].




  <cit.>
  There exists a constant c>0 such that
  
    sup_φ∈^3∖{ 0} qφ/φ_^3≥ cq_

  for all q∈.


The following theorem is called Korn's second inequality.


  <cit.>
  There exists a constant c>0 such that
  
    D(φ)_^3× 3≥ cφ_^3.

  for all φ∈^3.


The following embedding theorem is called Poincare's inequality.


  <cit.>
  There exists a constant c>0 such that 
  
    ∇φ_^3≥ cφ_
 
  for all φ∈ (i=12).


The following embedding theorem plays an important role in the proof of
the existence and the uniqueness of the solution to the Stokes problem
with the boundary condition (<ref>).





  <cit.>
  If Ω or Γ satisfy one of the following conditions;
  
    ΓC^1,1

  or
  
    Ω,

  then there exists a constant c>0 such that 
  
    ∇× v_^3≥ c v_^3
 
  for all v∈ H satisfying v=0.






 §.§ Weak formulations of (<ref>) and (<ref>)





We start by defining the weak solution to (<ref>).
Throughout of this paper, we suppose the following conditions;

    t^b∈ H^-1/2(Γ_2)^3,    
      F∈^3,


    g^b∈ H^-1/2(Γ_1),   
      p^b∈,   F∈.



  For u∈ H^2(Ω)^3,p∈ H^1(Ω) 
  and φ∈^3, 
  
    (-Δ u-∇( u)+∇ p)·φ
        =1/2 D(u):D(φ)- pφ
        -⟨ t,φ⟩_Γ_2

  holds, where t:=T_ν(u,p).




Proof.
We compute

    [                                                                                 (-Δ u-∇( u)+∇ p)·φ;                                                = -∑^3_i,j=1(u_i,jj+u_j,ij)φ_i
        +∑^3_i=1 p_,iφ_i; =∑^3_i,j=1{ (u_i,j+u_j,i)φ_i,j
         - (u_i,j+u_j,i)φ_i ν_j}
         +∑^3_i=1{- pφ_i,i
         + pφ_iν_i};           =1/2∑^3_i,j=1(u_i,j+u_j,i)(φ_i,j+φ_j,i)
         -∑^3_i=1 pφ_i,i
         -∑^3_i=1 T_ν(u,p)_iφ_i;                                                               =1/2 D(u):D(φ)- pφ
         -⟨ t,φ⟩_Γ_2, ]

which completes the proof.


For the second equation of (<ref>), taking ψ∈,  
we obtain

    [                       -( F)ψ                 = -(Δ p^PP)ψ;                              = -∂ p^PP/∂νψ
        +∇ p^PP·∇ψ;                                = -⟨ g^b,ψ⟩_Γ_1+∇ p^PP·∇ψ. ]

Therefore, the weak form of (<ref>) becomes as follows.
Find u^PP∈^3 and p^PP∈ such that

    PP'{[ 1/2 D(u^PP):D(φ)- p^PPφ                       =        F·φ-⟨ t^b,φ⟩_Γ_2                   φ∈^3,;               ∇ p^PP·∇ψ                       =     -( F)ψ+⟨ g^b,ψ⟩_Γ_1                     ψ∈,;                    p^PP                       =                 p^bΓ_2. ].



  If (u^PP,p^PP)∈^3× satisfies 
  u^PP∈ H^2(Ω)^3,p^PP∈ H^1(Ω) and (<ref>),
  then we have
  
    {[ (-Δ u^PP-∇( u^PP)+∇ p^PP-F)·φ                             =   ⟨ T_ν(u^PP,p^PP)-t^b,φ⟩_Γ_2                         φ∈^3,;                (-Δ p^PP+ F)·ψ                             =        ⟨-∂ p^PP/∂ν+g^b,ψ⟩_Γ_1                           ψ∈. ].

  Therefore, (u^PP,p^PP) satisfies (<ref>).



Next, we define the weak formulation of (<ref>).
For all φ∈^3, we obtain from the first equation of (<ref>),

    [                                F·φ
        =                         (-Δ u^S+∇ p^S)·φ;                                        =                 (-Δ u^S-∇( u^S)+∇ p^S)·φ;                                        = 1/2 D(u^S):D(φ)- p^Sφ
        -⟨ t^b,φ⟩_Γ_2. ]

Using this expression,
the weak form of the Stokes problem becomes as follows:
Find (u^S1,p^S1)∈^3× such that

    S1{[ 1/2 D(u^S1):D(φ)- p^S1φ                       =       F·φ -⟨ t^b,φ⟩_Γ_2                   φ∈^3,;                 -ψ u^S1                       =                       0                     ψ∈. ].



  If (u^S1,p^S1)∈^3× satisfies 
  u^S1∈ H^2(Ω)^3,p^S1∈ H^1(Ω) and (<ref>),
  then we have
  
    {[        (-Δ u^S1+∇ p^S1-F)·φ                           = ⟨ T_ν(u^S1,p^S1)-t^b,φ⟩_Γ_2                       φ∈^3,;                      ψ u^S1                           =                           0                         ψ∈.;                             ].

  Therefore, (u^S1,p^S1) satisfies (<ref>).






 §.§ Well-posedness of (<ref>), (<ref>)




We show the well-posedness of the problems (<ref>) and (<ref>)
in Theorem <ref> and <ref>.





Under the conditions (<ref>) and (<ref>),
there exists a unique solution (u^PP,p^PP)∈^3× satisfying (<ref>).





Proof.
From the second and third equations of (<ref>),
by using the Lax–Milgram theorem and Theorem <ref>,
p^PP∈ is uniquely determined.
Then u^PP∈^n is also uniquely determined 
from the first equation of (<ref>) by the Lax–Milgram theorem,
where the coercivity is guaranteed from Theorem <ref>.






Under the condition (<ref>),
there exists a unique solution (u^S1,p^S1)∈^3× satisfying (<ref>).





Proof.
By Theorems <ref> and <ref>,
the continuous bilinear form ^3×^3∋(u,φ)
↦ D(u):D(φ)∈ is coercive. 
By Theorems <ref> and <ref>, there exists a unique solution 
(u^S1,p^S1)∈^3× satisfying (<ref>).


We prove the following property of the solution to (<ref>).





If the weak solution (u^S1,p^S1)∈^3× to (<ref>)
satisfies p^S1∈ and Δ p^S1∈, then we have

    ∇ p^S1·∇ψ
      =-( F)ψ+⟨∂ p^S1/∂ν,ψ⟩_Γ_1

for all ψ∈.

  



Proof.
From the second equation of (<ref>) and u^S1∈, 
u^S1=0 holds in .
From the first equation of (<ref>), we obtain

    -Δ u^S1+∇ p^S1
      =-Δ u^S1-∇( u^S1)+∇ p^S1=F
        𝒟'(Ω).

Taking the divergence, we get

    F=(-Δ u^S1+∇ p^S1)=-Δ ( u^S1)+Δ p^S1=Δ p^S1  𝒟'(Ω).

By the assumptions Δ p^S1∈ and F∈,
Δ p^S1= F holds in .
Multiplying ψ∈ and integrating over Ω,
we get

    -( F)ψ=- (Δ p^S1)ψ
        =∇ p^S1·∇ψ
        -⟨∂ p^S1/∂ν,ψ⟩_Γ_1,

which is the desired result.




§ THE TRACTION BOUNDARY CONDITION


The purpose of this paper is to give an estimate of the difference
between the solutions of the Stokes problem and the pressure-Poisson problem.
Roughly speaking, from (<ref>) and the second equation of (<ref>),
Δ(p^S-p^PP)=0 holds. Hence, we get

    p^S-p^PP_≲(p^Sp^PPΓ),

where A≲ B means that there exists a constant c>0,
independent of A and B, such that
A≤ cB.
From (<ref>) and the second equation of (<ref>), we have

    -Δ(u^S-u^PP)=-∇(p^S-p^PP).

We obtain

    u^S-u^PP_^3≲∇(p^S-p^PP)_^3+(p^Sp^PPΓ).

Therefore, we have

    [                            u^S-u^PP_^3+ p^S-p^PP_;                        ≲ ((u^S,p^S)(u^PP,p^PP)Γ). ]

In other words, if we have a good prediction for the boundary data, 
then (<ref>) is good approximation for (<ref>).

In this section, we prove these types of estimates for the weak solutions.
Let the solutions of (<ref>) and (<ref>) be denoted by
(u^PP,p^PP) and (u^S1, p^S1), respectively.
First, we establish a lemma.


  If p∈, f∈ and g∈ H^-1/2(Γ_1) satisfy 
  
    ∇ p·∇ψ= fψ+⟨ g,ψ⟩_Γ_1  ψ∈,

  then there exists a constant c>0 such that
  
    p_≤ c(f_
          + g_H^-1/2(Γ_1)+ p_H^1/2(Γ_2)).






Proof.
Let p_0∈ such that p_0-p∈. 
Putting ψ:=p-p_0 in (<ref>), we have

    [                                ∇(p-p_0)^2_^3                                            =                           ∇ (p-p_0)·∇(p-p_0);                                                                                         =    f(p-p_0)+⟨ g,p-p_0⟩_Γ_2
      -∇ p_0·∇(p-p_0);                                                                                         ≤ f_ p-p_0_
      + g_H^-1/2(Γ_1) p-p_0_H^1/2(Γ_1);                                                                      +∇ p_0_^3∇(p-p_0)_^3;                                                                                         ≤     ( f_+c_1 g_H^-1/2(Γ_1)
      + p_0_) p-p_0_. ]

By Theorem <ref>, there exists a constant c_2>0 such that

    c_2 p-p_0^2_≤( f_+c_1 g_H^-1/2(Γ_1)+ p_0_) p-p_0_.

Hence,

    p-p_0_≤ c_3( f_+ g_H^-1/2(Γ_1)+ p_0_).

Since p_- p_0_≤ p-p_0_, we obtain

    p_≤ c_4( f_+ g_H^-1/2(Γ_1)+ p_0_).

For all p_0∈ satisfying p_0-p∈, (<ref>) holds.
Therefore,

    [                                         p_                                          ≤     c_4( f_+ g_H^-1/2(Γ_1)
      +inf_q∈ p+q_);                                                                                     =            c_4( f_+ g_H^-1/2(Γ_1)
      + p_/);                                                                                     ≤ c_5( f_+ g_H^-1/2(Γ_1)
      +  p_H^1/2(Γ_2)).;                                            ]



Using Lemma <ref>, we prove the following theorem
which is the main result of this section.


  If p^S1∈ and Δ p^S1∈, 
  there exists a constant c>0 such that
  
    u^S1-u^PP_^3+ p^S1-p^PP_≤ c(∂ p^S1/∂ν-g^b_H^-1/2(Γ_1)
          + p^S1-p^b_H^1/2(Γ_2)).






Proof.
Using Proposition <ref>, we obtain from (<ref>) and (<ref>),

    {[ 1/2 D(u^S1-u^PP):D(φ)                     =          (p^S1-p^PP)φ                 φ∈^3,;       ∇(p^S1-p^PP)·∇ψ                     = ⟨∂ p^S1/∂ν-g^b,ψ⟩_Γ_1                   ψ∈. ].

Putting φ:=u^S1-u^PP∈^3 in (<ref>), we get

    [     1/2 D(u^S1-u^PP)^2_^n× n     = (p^S1-p^PP)(u^S1-u^PP);                                   ≤p^S1-p^PP_(u^S1-u^PP)_;                              ≤√(3)p^S1-p^PP_u^S1-u^PP_^3. ]

From Theorem <ref>, 

    u^S1-u^PP_^3≤ c_1p^S1-p^PP_

holds for a constant c_1>0.
By the second equation of (<ref>) and Lemma <ref>,
there exists a constant c_2>0 such that

    [                                                p^S1-p^PP_                                                         ≤ c_2(∂ p^S1/∂ν-g^b_H^-1/2(Γ_1)
        + p^S1-p^PP_H^1/2(Γ_2));                                                                                                                   ≤ c_2(∂ p^S1/∂ν-g^b_H^-1/2(Γ_1)
        + p^S1-p^b_H^1/2(Γ_2)). ]

Therefore, it holds that

    u^S1-u^PP_^3+ p^S1-p^PP_≤ c_3(∂ p^S1/∂ν-g^b_H^-1/2(Γ_1)
        + p^S1-p^b_H^1/2(Γ_2)),

for a constant c_3>0.




§ BOUNDARY CONDITION INVOLVING PRESSURE


Let p^b∈.
We consider the Stokes problem with the boundary condition (<ref>):

    {[ -Δ u^S+∇ p^S            =            F           Ω,;          u^S            =            0           Ω,;            u            =            0         Γ_1,;          u×ν            =            0         Γ_2,;            p            =          p^b         Γ_2. ].

In this section, we evaluate the difference between the solutions to
(<ref>) and (<ref>) as in (<ref>).
First, we define the weak formulation of (<ref>)
and prove the existence and the uniqueness of the weak solution.
Next, we prove a proposition and a lemma as preparation 
for the proof of our main theorem: Theorem <ref>.

We define the weak formulation of (<ref>).
Multiplying the first equation of (<ref>) by v∈ H,
integrating by parts in Ω, and using the second equation of (<ref>),
we obtain

    F· v
      =(∇× u^S)·(∇× v)
      - p^S v+∫_Γ_2p^b v·ν,

where we have used the following lemma.

  For u∈ H^2(Ω)^3,p∈ H^1(Ω) and v∈ H, there holds
  
    (-Δ u+∇( u)+∇ p)· v
        =(∇× u)·(∇× v)
        - p v+∫_Γ_2 pv·ν.





Proof.
We compute

    [                                                                                                       (-Δ u+∇( u)+∇ p)· v;                                                            =                                            (∇×(∇× u)+∇ p)· v;                                                            =         (∇× u)·(∇× v)
          -((∇× u)×ν)· v
          - p v+ pv·ν;                                                            = (∇× u)·(∇× v)
          -∫_Γ(ν× v)·(∇× u)
          - p v+∫_Γ_2 pv·ν;                                                            =                        (∇× u)·(∇× v)
          - p v+∫_Γ_2 pv·ν. ]



The weak form of the Stokes problem (<ref>) becomes as follows: 
Find (u^S2,p^S2)∈ H× such that

    S2{[ (∇× u^S2)·(∇× v)- p^S2 v                        =        F· v-∫_Γ_2p^b v·ν                    v∈ H,;                  -ψ u^S2                        =                        0                      ψ∈. ].



  If (u^S2,p^S2)∈ H× satisfies 
  u^S2∈ H^2(Ω)^3,p^S2∈ H^1(Ω) and (<ref>),
  then we have
  
    {[ (-Δ u^S2+∇ p^S2-F)· v                     =   ∫_Γ_2(p^S2-p^b) v·ν                 v∈ H,;                ψ u^S2                     =                     0                   ψ∈. ].

  Therefore, (u^S2,p^S2) satisfies (<ref>).



We establish the well-posedness of this problem (<ref>) 
in the following theorem.





  <cit.>
  For F∈^3 and p^b∈,
  under the hypotheses of Theorem <ref>,
  there exists a unique solution (u^S2,p^S2)∈ H× to (<ref>).





Proof.
We set 

    a(u,v):=(∇× u)·(∇× v),  
      b(v,q):=- q v,  
      f(v):= F· v-∫_Γ_2p^b v·ν

for all u,v∈ H and q∈.
Clearly, a and b are continuous and bilinear forms and f∈ H^*. 
By Theorem <ref>, a is coercive on 
{ v∈ H | b(v,q)=0q∈}={ v∈ H |  v=0}.
By Theorem <ref>, b satisfies the assumption of Theorem <ref>.
Therefore, there exists a unique solution (u^S2,p^S2)∈ H× 
to (<ref>) by Theorem <ref>.


From here on, let the solutions of (<ref>) and (<ref>) be denoted by
(u^PP,p^PP) and (u^S2, p^S2), respectively.
The solution (u^S2, p^S2) to (<ref>) satisfies the following property.





If Δ u^S2+∇( u^S2)∈^3, p^S2∈ 
and Δ p^S2∈, then 

    {[     1/2 D(u^S2):D(φ)- p^S2φ                           = F·φ-⟨ T_ν(u^S2,p^S2),φ⟩_Γ_2                       φ∈^3,;                   ∇ p^S2·∇ψ                           =    -( F)ψ+⟨∂ p^S2/∂ν,ψ⟩_Γ_1                         ψ∈,;                        p^S2                           =                         p^b                        Γ_2. ].


  



Proof.
From the second equation of (<ref>) and u^S2∈, 
u^S2=0 holds in .
From the first equation of (<ref>), we obtain

    -Δ u^S2-∇( u^S2)+∇ p^S2
      =-Δ u^S2+∇( u^S2)+∇ p^S2=F

in 𝒟'(Ω). 
By the assumptions Δ u^S2+∇( u^S2)∈^3, 
p^S1∈ and F∈, equation (<ref>) holds in .
Multiplying φ∈ and integrating over Ω,
we get

    [                                                 F·φ                                                   =                         (-Δ u^S2-∇( u^S2)+∇ p^S2)·φ;                                                                                                       = 1/2 D(u^S2):D(φ)- p^S2φ
      +⟨ T_ν(u^S2,p^S2),φ⟩_Γ_2. ]

Taking the divergence of (<ref>), we have

    Δ p^S2= F   𝒟'(Ω).

By the assumptions Δ p^S2∈ and F∈,
Δ p^S2= F holds in .
Multiplying ψ∈ and integrating over Ω,
we get

    -( F)ψ=- (Δ p^S2)ψ
        =∇ p^S2·∇ψ
        -⟨∂ p^S2/∂ν,ψ⟩_Γ_1.

Multiplying (<ref>) by v∈ H and integrating over Ω,
we get

    [                                        F· v                                           =                (-Δ u^S2+∇( u^S2)+∇ p^S2)· v;                                                                                       = (∇× u^S2)·(∇× v)
      - p^S2 v+∫_Γ_2 p^S2 v·ν. ]

By the first equation of (<ref>), it holds that

    ∫_Γ_2p^S2 v·ν
      = -(∇× u^S2)·(∇× v)
      + p^S2 v+ F· v
      =∫_Γ_2p^b v·ν

for all v∈ H. Hence, p^S2=p^b holds in H^1/2(Γ_2).


We establish a lemma.


  If u∈^3, p∈ and t∈ H^-1/2(Γ_2) satisfy 
  
    1/2 D(u):D(φ)
        = pφ-⟨ t,φ⟩_Γ_2  φ∈,

  then there exists a constant c>0 such that
  
    u_^3≤ c(p_+ t_H^-1/2(Γ_2)).






Proof.
Putting φ:=u in (<ref>), we obtain

    [                      1/2D(u)^2_^3× 3                                    =                       p u-⟨ t,u⟩_Γ_2;                                                                         ≤ p_ u_
      + t_H^-1/2(Γ_2) u_H^1/2(Γ_2);                                                                         ≤    (√(3) p_+c_1 t_H^-1/2(Γ_2)) u_^3, ]

for a constant c_1>0.
By Theorem <ref>, there exists a constant c_2>0 such that

    c_2/2 u^2_^3≤(√(3) p_+c_1 t_H^-1/2(Γ_2)) u_^3.

Hence, we obtain the result with c=2/c_2max{√(3),c_1}.


The next theorem is the main result of this section.


  If Δ u^S2+∇( u^S2)∈^3, p^S2∈ 
  and Δ p^S2∈, then there exists a constant c>0 such that
  
    [                                                         p^S2-p^PP_                                                                  ≤                                        c∂ p^S2/∂ν-g^b_H^-1/2(Γ_1),;                                                       u^S2-u^PP_^3                                                                  ≤ c(
          ∂ p^S2/∂ν-g^b_H^-1/2(Γ_1)
          +t^S2-t^b_H^-1/2(Γ_2)^3), ]

  where t^S2=T_ν(u^S2,p^S2).





Proof.
Using Proposition <ref>, we obtain from (<ref>) and (<ref>),

    {[          1/2 D(u^S2-u^PP):D(φ)                              = (p^S2-p^PP)φ-⟨ t^S2-t^b,φ⟩_Γ_2                          φ∈^3,;                ∇(p^S2-p^PP)·∇ψ                              =          ⟨∂ p^S2/∂ν-g^b,ψ⟩_Γ_1                            ψ∈,;                      p^S2-p^PP                              =                              0                           Γ_2, ].

where t^S2=T_ν(u^S2,p^S2).
By the second equation of (<ref>) and Lemma <ref>,
there exists a constant c_1>0 such that

    [                                                p^S2-p^PP_                                                         ≤ c_1(∂ p^S2/∂ν-g^b_H^-1/2(Γ_1)
        + p^S2-p^PP_H^1/2(Γ_2));                                                                                                                   ≤                             c_1∂ p^S2/∂ν-g^b_H^-1/2(Γ_1). ]

By the first equation of (<ref>) and Lemma <ref>,

    [                                                   u^S2-u^PP_^3                                                              ≤                         c_2(p^S2-p^PP_+t^S2-t^b_H^-1/2(Γ_2)^3);                                                                                                                             ≤                         c_2(p^S2-p^PP_+t^S2-t^b_H^-1/2(Γ_2)^3);                                                                                                                             ≤ c_2(c_1∂ p^S2/∂ν-g^b_H^-1/2(Γ_1)
        +t^S2-t^b_H^-1/2(Γ_2)^3). ]





§ CONCLUSION AND FUTURE WORKS


We have proposed a new formulation for the pressure-Poisson problem (<ref>).
We have established error estimates between the solutions 
to (<ref>) and (<ref>) in Theorem <ref> and 
between the solutions to (<ref>) and (<ref>) in Theorem <ref>.
Theorem <ref> and <ref> state that 
if we have a good prediction for the boundary data (g^b or p^b),
then the pressure-Poisson problem is a good approximation for the Stokes problem.

For problem (<ref>), a finite element scheme is proposed in <cit.>
(under the assumption that Γ_2 is flat).
On the other hand, in many practical problems,
the projection method is more popular due to its easiness in implementation.
Numerical comparison of (<ref>) and (<ref>) is 
one of our interesting future works from those points of view.

As another extension of our research,
generalization of our results to the Navier–Stokes problem
is important but is still completely open.







