Linear transport equation with Lipschitz conditions Given the equation here, I would like to ask the following relaxed question:
Consider the PDE
$$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$
with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$
I am wondering then if $q$ is Lipschitz and $p$ is Schwartz as $f_0$, does there exist a solution $(t,x)\mapsto f(t,x)$ for this equation that decays faster than any polynomial in the space variable $x$ for any fixed time $t>0$?
 A: The anwer is yes.
Preliminary observations:
Let us first consider the case $p=0$.
Since $q$ is globally Lipschitz on $\mathbb{R}^n$, say with Lipschitz constant $L$, all solutions of the ODE $\dot x = q(x)$ on $\mathbb{R}^n$ exist globally and, if $(\varphi_t)_{t \in \mathbb{R}}$ denotes the induced flow in $\mathbb{R}^n$, each function $\varphi_t: \mathbb{R}^n \to \mathbb{R}^n$ is Lipschitz continuous with constant $e^{|t|L}$ (this follows from Grönwall's lemma).
Since we assumed $p=0$, the solution of your PDE is given by
$$
  f(t,x) = f_0(\varphi_t(x)) \tag{*}\label{1}
$$
for each $t$ and $x$.
Hence, if $f_0$ decays faster than every polynomial, so does $f(t, \cdot)$ for each fixed time $t$.
Let us now phrase this in a more functional analytic language: For each integer $k \ge 0$, consinder the weight function $w_k: \mathbb{R}^n \to [1,\infty)$ given by $w_k(x) =1 + \lvert x \rvert^k$, and consider the Banach space $E_k$ of all continuous functions $f: \mathbb{R}^n \to \mathbb{R}$ which satisfies $w_k f \in C_0(\mathbb{R}^n; \mathbb{R})$; we endow the space $E_k$ with the weighted supremum norm $\|\cdot\|_{w_k}$ given by
$$
  \|f\|_{w_k} = \|w_k f\|_\infty.
$$
It follows from $\eqref{1}$ that the Koopman group $(T_k(t))_{t \in \mathbb{R}}$ given by
$$
  T_k(t)f = f \circ \varphi_t
$$
for each $f \in E_k$ is a well-defined one-parameter group of bounded linear operators on $E_k$. Moreover, the orbits of the group are continuous with respect to the weak topology on $E_k$ (use that (i) $E_k$ is isomorphic to $C_0(\mathbb{R}^n; \mathbb{R})$ via multiplication with $w_k$, that (ii) for each $x \in \mathbb{R}^n$, the mapping $t \mapsto \varphi_t(x)$ is continuous, and (iii) the Riesz representation theorem for measures on $C_0(\mathbb{R}^n; \mathbb{R})$).
Hence, by a standard result from $C_0$-semigroup theory, the the group $(T_k(t))_{t \in \mathbb{R}}$ is even continuous with respect to the strong operator topology, i.e., it is a $C_0$-group on $E_k$.
It is trivial, but important to observe, that the groups $(T_k(t))_{t \in \mathbb{R}}$ act consistently on the spaces $E_k$.
Using these preliminary observations, we can show:
Theorem. Let $p$ be continuous and decay faster than every polynomial.
(i) For each $f_0 \in C_0(\mathbb{R}^n; \mathbb{R})$ there exists a unique mild solution $f: \mathbb{R} \to C_0(\mathbb{R}^n; \mathbb{R})$ of the Cauchy problem
$$
  \dot f = \langle q, \nabla \rangle f(t) + p, \qquad f(0) = f_0.
$$
(ii) If $f_0$ decays faster than every polynomial and $f: \mathbb{R} \to C_0(\mathbb{R}^n; \mathbb{R})$ denotes the mild solution from (i), then for each fixed time $t \in \mathbb{R}$, the function $f(t)$ also decays faster (in space) than every polynomial.
Proof.
Fix $k \ge 0$. Since $(T_k(t))$ is a $C_0$-group on $E_k$ and since $p \in E_k$, it follows that, if $f_0 \in E_k$, the PDE has a unique mild solution $f: [0,\infty) \to E_k$ given by
$$
  f(t) = T_k(t)f_0 + \int_0^t T_k(s) p \, ds,
$$
where the integral is a Bochner integral in $E_k$. Applying this to $k=0$, we obtain (i).
Moreover, these solutions are consistent on the $E_k$-spaces (i.e., if $f_0 \in E_{k+1} \subseteq E_k$, then the mild solutions in $E_k$ and $E_{k+1}$ are the same). If $f_0$ decays faster than every polynomial, then we have $f_0 \in E_k$ for each $k$. Hence, for each time $t$ and each $k$, we have $f(t) \in E_k$. Thus, for each time $t$, the function $f(t)$ decays faster in space than any polynomial. $\square$
