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In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A 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:

    $$ \begin{align} -\nabla P+\mu \nabla^2 u+\rho b&=0\\ \nabla \cdot u &=0 \end{align} $$

  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?
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Yes, p is the pressure in the Stokes equation. Yes, the Green's function given here is for representing the velocity. Why is its form strange, and what is "the most general case"? – Michael Renardy Jun 3 '11 at 17:34
@Michael Renardy: Thanks for the comment. Actually I don't understand how can one get $\frac{1}{8\pi\mu}$ in the solution formula. – Jack Jun 3 '11 at 17:46
I think it is just a matter of how you define $G_{ij}$. – Michael Renardy Jun 3 '11 at 18:57
up vote 1 down vote accepted

I moved this question to math.SE 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 to the question. Instead of closing or deleting the question, I think it's worth putting the link here.

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