Is it possible to identify the vector field $\vec{\mathbf{F}}$ from only knowing the divergence $\nabla \cdot \vec{\mathbf{F}}$?

Question

Say there is some unknown vector field $ \vec{\mathbf{x}}(x, y) $ we are trying to find from only knowing it’s divergence $ u(x, y) + v(x, y) $ where $u$ and $v$ are the partial derivatives of each part, as described by the formula for divergence.

Is this possible? I guess we would first of all need to find the gradient, and then partially integrate each part? How would that work?

Let’s say the divergence of a field would be $4x + 2y$ for a simple example. This is the same as saying $\frac{\partial}{\partial x}r + \frac{\partial}{\partial y}t = 4x + 2y$, where $r$ and $t$ are multivariable functions of $x$ and $y$.

I’m pretty sure this is some PDE. As my knowledge of partial differential equations is ZERO, I cannot help myself here. My general goal here is to construct one single formula in the form of a differential equation which would allow us to find the vector field from the divergence.

Hence

Any help is appreciated!

Answer 1

One possible avenue of attack is by solving the Laplace equation. Say you want to find $F:\Omega\rightarrow \mathbb{R}^n$ while $\Omega\subseteq\mathbb{R}^n$ is some open set such that $$\operatorname{div} F = f$$ for some $f:\Omega\rightarrow\mathbb{R}$. You can make an ansatz by saying, that $F$ is the gradient of a function $u$, i.e. $$F=\nabla u.$$ Then you can plug this into your original equation and obtain $$\operatorname{div}\nabla u = \Delta u = f.$$ Depending on $\Omega$ you might be able to find an exact solution formula. For the ball, half spaces and the whole space itself they exist. The key word you are looking for would be Greens function/representation or fundamental solution. Typical references would be Gilbarg and Trudingers book about Elliptic partial differential equations or Evans Introduction to partial differential equations or https://en.wikipedia.org/wiki/Green%27s_function_for_the_three-variable_Laplace_equation

The Laplace equation can also be solved numerically by e.g. finite element methods etc.

Oh and before I forget, by Egor Ivanovs comment you cannot expect uniqueness.

Answer 2

In general, finding a vector field is impossible if one only knows its divergence. It is the consequence of the Helmholtz-Hodge theorem, which says that a smooth vector field $F$, defined on a bounded or an unbounded domain, can be uniquely decomposed into three components: 1) an irrotational component, which is normal to the boundary; 2) an incompressible component which is parallel to the boundary; and 3) a harmonic component.