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I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple.

  1. Can some one tell me some reference to study about the invertibility of Divergence operator $\operatorname{div}\colon C^1(\omega)\to G$ where $G$ is a space of real valued function and $\omega$ is a subset of $\Bbb R^2$. Here I assume a Dirichlet type condition on the boundary of $\omega$ is specified and all boundary and domain have nice smoothness.
  2. In above context can someone give me some reference on the Null space structure of the divergence operator operating on differentiable maps defined on $\Bbb R^2$ ?

Ariwn

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"curl"? There is only such a thing (as a map from vector fields to vector fields) in $n=3$. –  Robert Israel Sep 6 '12 at 19:36
    
By curl you mean the exterior derivative acting on 1-form? And why should $G$ be a space of vector valued function? –  Shuhao Cao Sep 6 '12 at 19:40
    
Divergence takes vector fields to scalar fields. –  Robert Israel Sep 6 '12 at 19:50
    
Sorry for my typo error and horrible english.I mean it maps vector valued maps to scalar valued and the curl of a map case in question 2 was for $R^3 (poincare type )$ but , here I need in $R^2$ –  user39646 Sep 6 '12 at 19:53
    
Perhaps you want the analogue of Helmholtz decomposition in two dimensions. I think this is dealt with in fluid mechanics in terms of the stream function and velocity potential, but I'm afraid I don't have a reference handy. –  Rahul Sep 6 '12 at 20:13

2 Answers 2

A divergence-free vector field corresponds to an incompressible flow. The null space is infinite-dimensional. For example, to get a divergence-free vector field supported in a ball $\{|x| \le r\}$, you can take something like this: $$V(x) = (f(|x|) x_2, -f(|x|) x_1, 0, \ldots, 0)$$ where $f$ is a smooth function supported in $[0,r]$.

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Thank you for the reference.In general I need that if $\nabla \cdot q_1 = \nabla \cdot q_2 $ on $\Omega $ in $R^2$ then what I can conclude in general about the relationships of $q_1$ and $q_2$.To be more specific I need to find $q_1 \cdot e_i $ if $\nabla \cdot q_1 = 0 $ where $e_i$ is the unit normal vector in $R^2 $. –  user39646 Sep 6 '12 at 20:03
    
It could be any smooth function (subject to boundary conditions). That is, given $v_1$ you can take $v_2$ to be a solution of $\dfrac{\partial v_2}{\partial x_2} = - \dfrac{\partial v_1}{\partial x_1}$ and have $\nabla \cdot (v_1, v_2) = 0$. –  Robert Israel Sep 6 '12 at 21:14

Like Robert pointed out, this problem is related to Stokes problem, a limiting case of the incompressible Navier-Stokes equation.

For Arbitrary dimension as you first asked, you could refer to this paper, in $\mathbb{R}^3$, the existence of a vector potential to solve the divergence equation can be found here.

For $\mathbb{R}^2$, I am looking at the book by Girault and Raviart now, the complete results of the function spaces related to divergence operator can be found from page 22, and they are like an application of De Rham's work in cohomology. $\newcommand{\v}[1]{\boldsymbol{#1}}$ Basic idea is for any smooth vector fields $\v{v}$ in $\mathbb{R}^2$, there exists a decomposition $$ \v{v} = \mathbf{grad} \phi + \mathbf{curl} \psi $$ if the boundary of domain of interest has certain smoothness(say Lipschitz). Say if your problem is to find $\v{v}$ such that $$ \left\{ \begin{aligned} \mathrm{div} \v{v} &= f \text{ in } \Omega \\ \v{v} &= \v{g} \text{ on } \partial\Omega \end{aligned} \right. $$ Plugging the decomposition would enable you to establish a Neumann problem of Laplace equation for $\phi$, and any $\psi \in H^1$ that has a zero boundary condition would satisfy $\mathbf{curl} \psi \cdot \v{n} = 0$ on boundary.

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