Consider two embeddings of the $2$-sphere in $R^3$ -- call them $S_0$ and $S_1$. My question is whether there is always a flow on $R^3$ that takes $S_0$ to $S_1$?
By my understanding, one would need to assume that the embeddings are bicollared, so as to avoid wild embeddings, such as the Alexander horned sphere, but maybe it is sufficient to assume that they are smooth embeddings (?). Assuming this, we know that there is an isomorphism of $R^3$ that takes $S_0$ to $S_1$, by the Generalized Schoenflies Theorem (Morton Brown). Then, according to a theorem of Kirby, this can be achieved by an isotopy.
Assuming that the embeddings are smooth, and the isotopy is smooth (can we assume this?), then can one strengthen this result to assert that there will be a smooth flow that takes the first sphere to the second? Differentiating the isotopy would mean that there is a flow along a time-varying vector field which takes $S_0$ at time $t=0$ to $S_1$ at time $t=1$. However, I am interested in whether there will always be a flow on a time-invariant vector field that will to this. Another way of stating this is that there is a smooth $1$-parameter group of diffeomorphisms of $R^3$ that will take $S_0$ to $S_1$ at times $t=0$ and $t=1$.
Intuitively, this will always be true, I imagine.
One could obviously generalize this question to higher dimensions, but for now I am mainly interested in this low-dimensional case.