This question was originally posted by me on math.stackexchange but I didn't get any answers and I thought that perhaps it would be better off here. I hope it's appropriate, I've encountered the problem while reading research articles but it's still possible that the theorem I'm asking for is actually elementary. I'm trying to work my way through some articles on global existence of solutions to vlasov equation and I'm having trouble coming up as to why a $C^1$-norm bound on our density function contradicts the maximal time interval being bounded. Anyway, here we go:
I'm looking for references to results regarding maximal time existence of solutions of a certain transport-like PDE, more precisely this one (I'm working in three dimensional space, that is $x$ and $v$ are three dimensional if it matters): $$ \partial_t f + v \cdot \nabla_x f + E(f,t,x) \cdot \nabla_v f = 0$$ where the important fact is that $E$ depends on $f$, on it's integral with respect to $v$ to be exact. I'm not very experienced with nonlinear equations but I was told there was a general theorem for situations like this one saying, that if the maximal interval of existence $[0, T)$ is bounded, i.e. $T<\infty$ and the solution cannot be extended further than $T$ then its $C^1$ norm must tend to $\infty$ when $t \rightarrow T$. I would very much appreciate any pointers as to sources in which I can look for a result like this one, to anything relevant in fact, thanks!