Consider $\mathbb{R}\_t \times \mathbb{R}\_x ^n$ , let $b\_1(t,x)$ and $b\_2 (t,x)$ be two velocity fields with all the regularity you want and consider the flow of the point $(0,x\_0)$ for a time $T$. Now, I can view the trajectories of $(0,x\_0)$ as curves in $\mathbb{R}^{n+1}$ like $ \gamma\_1 =(t, \Phi\_1 (t,x\_0))$ and $\gamma \_2= (t, \Phi\_2 (t,x\_0))$ with $\Phi\_i$ the flux relative to $b\_i$. The question is: if I consider $\tilde{\gamma} =\gamma\_1 \cup \gamma\_2 \cup I\_T$ where $I\_T$ is a segment in $\mathbb{R}^{n+1}$ joining $(T,\Phi\_1 (T, x\_0) )$ and $(T, \Phi\_2 (T, x\_0))$ , I can consider the Plateau problem associated to the rectifiable set $\tilde{\gamma}$, and by the general theory I'll have a minimal surface whose boundary is $\tilde{\gamma}$. The question is: given how my $\gamma$ is constructed, is there a way to estimate the area of such surface in terms of $b\_1$ and $b\_2$?


EDIT:
Actually, what I want is to integrate such an estimate, starting from a certain mass distribution at time 0, so I would like something whixh takes that into account.



Plateau problem for fluxes of curves
Almgren (<https://mathscinet.ams.org/mathscinet-getitem?mr=855173>) proves that the Plateau solution will satisfy the 2-dimensional Euclidean isoperimetric inequality. So you can bound
$$
\textrm{Area(Plateau Solution)} \leq (4\pi)^{-1}(L(\gamma\_1)+L(\gamma\_2)+L(I\_T))^2.
$$
If you just want a coarse estimate you should be able to bound each term by the $L^\infty$ norm of $b\_1,b\_2$.


Of course, this inequality is unlikely to be close to optimal. (For example, if $b\_1=b\_2$ then the area of the Plateau solution is $=0$ while the RHS above is nonzero). If you want an estimate that works better in this case you might construct the explicit competitor
$$
\Gamma : = \bigcup\_{t=0}^T I\_{\Phi\_1(t,x\_0),\Phi\_2(t,x\_1)} \times \{t\}
$$
namely you union up all the lines between the points in the flow. You should be able to compute the area of this in terms of $b\_1,b\_2$ using the co-area formula.


EDIT: As requested here is a formula for the area of $\Gamma$. I did it via a parametrization instead of co-area but I think co-area would also work.


I parametrize $\Gamma$ by $F: [0,1]\times [0,T]\to\mathbb{R}^{n+1}$
$$
F(s,t) : = (s\Phi\_1(t)+(1-s)\Phi\_2(t),t)
$$
(dropping the $x\_0$ notation). Then we have
$$
\partial\_s F = \Phi\_1-\Phi\_2, \qquad \partial\_t F = s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2 + \mathbf{e}\_{n+1}.
$$
In particular, the induced metric is
\begin{align\*}
g\_{ss} & = |\Phi\_1-\Phi\_2|^2 \\
g\_{st} & = \langle \Phi\_1-\Phi\_2,s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2\rangle\\
g\_{tt} & = 1 + |s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2|^2.
\end{align\*}
Hence
$$
\det g = (1 + |s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2|^2)|\Phi\_1-\Phi\_2|^2 - \langle \Phi\_1-\Phi\_2,s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2\rangle^2. 
$$
This gives the explicit formula
$$
\textrm{area}(\Gamma) = \int\_{[0,1]\times [0,T]} \sqrt{(1 + |s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2|^2)|\Phi\_1-\Phi\_2|^2 - \langle \Phi\_1-\Phi\_2,s\partial\_t\Phi\_1 +(1-s)\partial\_t \Phi\_2\rangle^2}. 
$$
If you want to write this in terms of $b\_1$ and $b\_2$ you can do the following (perhaps one can be more precise here to get a better estimate). First of all
$$
|\partial\_t \Phi\_i| = |b\_i| \leq \Vert b\_i\Vert\_{C^0}
$$
Secondly, by the argument [here](https://mathoverflow.net/a/300284/1540) we can bound
$$
|\Phi\_1-\Phi\_2| \leq L^{-1}(e^{Lt}-1)|b\_1-b\_2|\_{L^\infty}.
$$
for $L$ the Lipschitz constant of one of the vector fields.


Using this above we should get
$$
\textrm{area}(\Gamma) \leq (1 + (|b\_1|\_{C^0}+|b\_2|\_{C^0}) |b\_1-b\_2|\_{C^0}^2 L^{-2}(e^{LT}-LT-1). 
$$


Note that I discarded the $g\_{st}$ term, which could improve this estimate if you had some way of knowing it was big. But it does have the nice feature that it vanishes if $b\_1=b\_2$ and it's $O(T^2)$ for $T$ small.


I'm sure you can improve the estimate used above in the case of large $T$ (e.g. if $b\_i$ are bounded, then the flows should diverge at most linearly).


