Can we approximate a vector field on the plane with non-vanishing vector fields in $L^2$?

I'll assume the zeroes of $V$ are isolated on an open neighborhood of the unit disk.

Here is a procedure for approximating $V$ in $L^2$ by a vector field in the same class - $C^\infty_c$ with isolated zeros in a neighborhood of the unit disk - but with one fewer zero in $\mathbb D^2.$ The idea is to push the zero out. (This is basically what I meant in the linked answer by "composing with a suitable diffeo".)

Pick points $(x_0,y_0)$ and $(x_1,y_1)$ such that:

• $(x_0,y_0)\in\mathbb D^2.$
• $(x_1,y_1)\not\in\mathbb D^2.$
• $V(x_0,y_0)=(0,0).$
• The straight line segment from $(x_0,y_0)$ to $(x_1,y_1)$ contains no zeroes of $V$ except $(x_0,y_0).$

Let $C$ be the straight line segment from $(x_0,y_0)$ to $(x_1,y_1).$ Let $C_n$ be a sequence of open neighborhoods of $C$ such that $$\mu(C_n)\to 0$$ where $\mu$ is Lebesgue measure, and such that $C_n$ contains no zeros of $V$ except $(x_0,y_0).$ Using Whitney's extension theorem pick a function $s:\mathbb R^2\to\mathbb R$ such that:

• $s(x,y)=1$ for $(x,y)\in C$
• $s(x,y)=0$ for $(x,y)\not\in C_n$

Since $s$ is compactly supported, $(x,y)\mapsto s(x,y)(x_1-x_0,y_1-y_0)$ is a complete vector field and defines a flow globally: there are diffeomorphisms $\psi_t$ for $t\in\mathbb R$ where $\psi_0$ is the identity and $\frac{d}{dt}\psi_t(x,y)=s(x,y)(x_1-x_0,y_1-y_0).$ Define $$V_n=V \circ \psi_{-1}.$$ Since $$\|V-V_n\|_2^2\leq \mu(C_n)(2\max|V|)^2$$ we have $V_n\to V$ in $L^2.$

For $(x,y)\in C_n$ we have $V_n(x,y)=(0,0)$ if and only if $\psi_{-1}(x,y)=(x_0,y_0),$ which is equivalent to $(x,y)=(x_1,y_1).$ And outside $C_n,$ the vector fields $V_n$ and $V$ are the same. So the number of zeros inside the unit disk has decreased by one.