In mathematics, a [Green's function](http://en.wikipedia.org/wiki/Green%27s_function) is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A [fundamental solution](http://en.wikipedia.org/wiki/Fundamental_solution) for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function.



> 
> In C. POZRIKIDIS's *Boundary Integral
>  and Singularity Methods for Linearized
>  Viscous Flow*, 
> 
> 
> The Green's functions of Stokes flow
>  represent solutions of the continuity
>  equation $\nabla\cdot {\bf u}=0$ and the
>  singularly forced Stokes equation
>  $$-\nabla P+\mu
> \nabla^2{\bf u}+{\bf g}\delta({\bf x-x\_0})=0 $$
> 
> 
> where ${\bf g}$ is an arbitrary constant,
>  ${\bf x\_0}$ is an arbitrary point, and
>  $\delta$ is the three-dimensional
>  delta function. Introducing the Green's function ${\bf G}$, we write the solution of (2.1.1) 
>  in the form
>  $$u\_i({\bf x})=\frac{1}{8\pi\mu}G\_{ij}({\bf x,x\_0})g\_j$$
> 
> 
> 


I am confused with the Green's function in this text. 


Here are my **questions**:


1. Is $P({\bf x})$ supposed to be the unknown
in the [Stokes equations](http://en.wikipedia.org/wiki/Stokes_flow#Stokes_equations-):


$$ \begin{align}
-\nabla P+\mu \nabla^2 u+\rho b&=0\\
\nabla \cdot u &=0
\end{align}
$$
2. What does the Green's function mean here? (Is it "with respect to" $u$?)  ~~Why is it of that strange form?~~ Why is the solution of this kind of form?
3. ~~How can one get $\frac{1}{8\pi\mu}$?~~What is the relation between ${\bf G}$ and $G\_{ij}$? As I understand, $G\_{ij}$ are the components and ${\bf G}:{\mathbb R}^3\to{\mathbb R}^3$. Then one should write:
$${\bf G}({\bf x})=\begin{bmatrix}
G\_1({\bf x})\\ G\_2({\bf x})\\ G\_3({\bf x})\end{bmatrix}$$
where $G\_i:{\mathbb R}^3\to{\mathbb R}$. What is $G\_{ij}$?
4. What's the Green's function in the most general case?



Green's functions of Stokes flow?
I moved this question to [math.SE](https://math.stackexchange.com/q/42714/9464) a month ago. This is indeed the problem I got from the research, though it may not very appropriate here. 


@Willie Wong gave a very nice [answer](https://math.stackexchange.com/questions/42714/greens-functions-of-stokes-flow/50398#50398) to the question. Instead of closing or deleting the question, I think it's worth putting the link here.


