Consider the initial value problem for the so-called Liouville equation

$$\begin{gathered} f_{t}+v \cdot \nabla_{x} f-\nabla V(x) \cdot \nabla_{v} f=0,(x, v) \in \mathbb{R}^{2 d}, t \in \mathbb{R} \\ f(x, v, t=0)=f_{I}(x, v) \end{gathered}$$

for the function $f=f(x, v, t)$ on $\mathbb{R}^{2 d} \times \mathbb{R}$. Assume that $V=V(x)$ is a given function with $V, \nabla_{x} V$ Lipschitz continuous on $\mathbb{R}^{d}$.

(i) Let $f_{I}(x, v)=\delta\left(x-x_{0}, v-v_{0}\right)$, for $x_{0}, v_{0} \in \mathbb{R}^{d}$ given. Show that a solution $f$ is given by

$$f(x, v, t)=\delta\left(x-\hat{x}\left(t, x_{0}, v_{0}\right), v-\hat{v}\left(t, x_{0}, v_{0}\right)\right)$$

where $(\hat{x}, \hat{v})$ solve the Newtonian system

$$\begin{array}{ll} \dot{\hat{x}}=\hat{v}, & \hat{x}(t=0)=x_{0} \\ \dot{\hat{v}}=-\nabla V(\hat{x}), & \hat{v}(t=0)=v_{0} \end{array}$$

(ii) Let $f_{I} \in L_{l o c}^{1}\left(\mathbb{R}^{2 d}\right), f_{I} \geqslant 0$. Prove (by using characteristics) that $f$ remains nonnegative (as long as it exists).

(iii) Let $f_{I} \in L^{p}\left(\mathbb{R}^{2 d}\right), f_{I} \geqslant 0$ on $\mathbb{R}^{2 d}$. Show (by a formal argument) that

$$\|f(\cdot, \cdot, t)\|_{L^{p}\left(\mathbb{R}^{2 d}\right)}=\left\|f_{I}\right\|_{L^{p}\left(\mathbb{R}^{2 d}\right)}$$

for all $t \in \mathbb{R}, 1 \leqslant p<\infty$.

(iv) Let $V(x)=\frac{1}{2}|x|^{2}$. Use the method of characteristics to solve the initial value problem for general initial data.