Suppose there is a vector field $F_1$ with two sets of integral curves glued together at their common boundary line of $x^1+y^1=1,$ where the integral curves are of the forms:
$$\zeta:= \{ (x, y) \in \Bbb R^2 | x^s + y^s = 1 \},$$ $$\tau:= \{ (x, y) \in \Bbb R^2 | (1-x)^s + (1-y)^s = 1 \}.$$
for $ x,y\in \Re(0,1)$ and $1 \le \Re(s) < \infty. $
Reflect a congruent copy of each integral curve above about $x=1/2$ and denote a new vector field $F_2.$ Define $F_1+F_2=F_3.$ Let the sources of the flows $F_1$ and $F_2$ start at $(0,0)$ and $(1,0).$
Consider all diffeomorphisms of the mesh produced by the integral curves of $F_1$ and $F_2$ from the unit square to the unit square.
Does the collision of $F_1$ and $F_2$ ever produce a smooth $F_3$ for any initial velocity?
If the initial velocity is large enough, it will cause the blowup of $F_3.$ I'm wondering if there is a tight lower bound on the velocities of $F_1$ and $F_2$ that, when added, produce a smooth $F_3,$ or not.