The specific PDE I am interested in is from classical mechanics: consider the phase space distribution function $\rho(\vec q,\vec p,t)$, which satisfies (by Liouville's theorem): $$\partial_t \rho(\vec q,\vec p,t) = \sum_{i=1}^N \frac{\partial H(\vec q,\vec p)}{\partial p_i}\partial_{q_i}\rho(\vec q,\vec p,t) - \frac{\partial H(\vec q,\vec p)}{\partial q_i}\partial_{p_i}\rho(\vec q,\vec p,t)$$ (Note that this is a linear PDE)
Suppose $H(\vec q,\vec p)$ is such that $\rho(\vec q,\vec p,t)=0$ for $|q_i|>L$ and $|p_i|>P$ for all $t$ (for example, suppose the system is inside a box of length $L$).
My naive question is: if the initial condition $\rho(\vec q,\vec p,0)$ is "nice", will $\rho(\vec q,\vec p,t)$ remain "nice" for $t>0$.
My attempt at making this more precise is: Suppose the initial condition $\rho(\vec q,\vec p,0)$ is analytic and suppose its fourier series converges pointwise. Will $\rho(\vec q,\vec p,t)$ remain analytic for $t>0$? Will its fourier series converge pointwise for $t>0$?
The typical way of solving first order equations like this is using the method of characteristics. The idea is to go back to the ODEs describing particle trajectories, which are effectively Hamilton's equations.
Precisely, we look for curves $P(s,t,p,q),Q(s,t,p,q)\in\mathbb{R}^N$ satisfying $P(t,t,p,q)=p$, $Q(t,t,p,q)=q$ along which the phase space distribution is conserved, i.e.
$$\frac{d}{ds}\rho(Q(s,t,p,q),P(s,t,p,q),s) = 0$$
If we can find such curves, then the solution is given explicitly by $$\rho(q,p,t) = \rho_0(Q(0,t,x,v),P(0,t,x,v)) \tag{*}$$ where $\rho_0 = \rho|_{t=0}$, since if the distribution along trajectories is conserved, then its value at $s=t$ is the same as its value at $s=0$, which correspond to the left and right hand sides of (*) respectively.
But the chain rule yields
$$\frac{d}{ds}\rho(Q(s,t,p,q),P(s,t,p,q),s) = (\partial_s\rho+\dot{Q}\cdot\nabla_q\rho+\dot{P}\cdot\nabla_p\rho)(Q(s,t,p,q),P(s,t,p,q),s) = 0$$
So, comparing to your original PDE we obtain the ODEs
\begin{gather} \tfrac{d}{ds}Q(s,t,p,q) = -\nabla_p H(Q(s,t,p,q),P(s,t,p,q)), \ \ \ Q(t,t,p,q) = q\\ \tfrac{d}{ds}P(s,t,p,q) = \nabla_q H(Q(s,t,p,q),P(s,t,p,q)), \ \ \ P(t,t,p,q) = p \end{gather}
From classical theory of ODEs, if $H$ is analytic, then so are $P,Q$. Then by the chain rule and (*), if $P,Q, \rho_0$ are smooth, then so will $\rho$. Analyticity might be a little trickier, but it should follow from the fact that compositions of (real) analytic functions are (real) analytic.