Consider a vector field $f\colon \mathbb{R}^2 \to \mathbb{R}^2$ defined as $$ (x_1, x_2) \mapsto f(x_1,x_2) = \left(\frac{\alpha}{\alpha+x_2^m}-\beta x_1,\frac{\alpha}{\alpha+x_1^m}-\beta x_2\right) $$ where $\alpha$, $\beta$ are real positive numbers and $m>1$ is an integer number.
My question. The vector field $f$ is clearly not conservative. However, is it possible to find a "good approximation" of $f$ that is conservative?
For "good approximation" I mean any vector field $\tilde f$ that is close to $f$ with respect to some metric.
Any comment is welcome. Thanks!
This doesn't work for your particular $\mathbf{F}$, but under suitable regularity and growth assumptions, the minimizer of $$\frac{1}{2}\int \lvert\boldsymbol{\nabla}\phi(\mathbf{x}')+\mathbf{F}(\mathbf{x'})\rvert^2\mathrm{d}V_{\mathbf{x'}}$$
is the solenoidal component of the Helmholtz decomposition of $\mathbf{F}$
$$\begin{align}\phi(\mathbf{x})&=\int G(\mathbf{x}-\mathbf{x'})\boldsymbol{\nabla}\cdot\mathbf{F}(\mathbf{x}')\mathrm{d}V_{\mathbf{x}'}\\ -\nabla^2G(\mathbf{x}-\mathbf{x}')&=\delta(\mathbf{x}-\mathbf{x}')\text{.} \end{align}$$ We can interpret $-\boldsymbol{\nabla}\phi$ as an $L^2$ "projection" of $\mathbf{F}$ onto the linear space of conservative vector fields.