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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!

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    $\begingroup$ Viewing the PDE as an abstract ODE in Banach spaces can give a hint/flavour compared to the following usual result for finite dimensional ODE's: if $\frac{dy}{dt}=F(t,y)$ for a 'nice' $F$, then the only possibility for the maximal existence time $T$ to be finite is $\lim\limits_{t\to T^-}|y(t)|=\infty$. Then adapting to infinite dimensional PDE's usually requires to find a suitable functional setting (you mentioned the $C^1$ norm, but weaker Sobolev spaces are also very frequent). Apart from that I'm no expert in transport equations so I don't have a precise reference to point you to, sorry. $\endgroup$
    – leo monsaingeon
    Commented May 11, 2014 at 18:26

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What you are looking for is a breakdown or continuation criterion.

For the case of Vlasov, it is not too difficult, since it is just a nonlinear transport equation, which reduces to basically a nonlinear ODE once you use the method of characteristics; and the continuation criterion is essentially the same as the proof of local existence of solutions. Since you asked for a reference, the linear theory can be found in Mouhot's lecture notes (see Chapter 1 and 3). The nonlinear theory you do an iteration and use smallness from the small time interval to get a contraction.

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