Suppose the conditions:
$1)$ $p \in [1,+\infty]$ ,
$2)$ $u_{0}(x) \in L^{p}(\mathbb{R^{N}})$,
$3)$ $c + \mbox{div} b \in L^{1}([0,T];L^{q}_{\mbox{loc}}(\mathbb{R^{N}}))$, $b \in L^{1}([0,T];(L^{q}_{\mbox{loc}}(\mathbb{R^{N}}))^{N})$, where $q$ is the conjugate exponent of $p$,
$4)$ $b \in L^{1}([0,T];(L^{1}_{\mbox{loc}}(\mathbb{R^{N}}))^{N})$ and $c \in L^{1}([0,T];(L^{1}_{\mbox{loc}}(\mathbb{R^{N}}))^{N})$,
$5)$ If $p > 1$, assume $c + \frac{1}{p} \mbox{div} b \in L^{1}([0,T]; L^{\infty}(\mathbb{R^{N}}))$
$6)$ If $p = 1$, assume $c$, $\mbox{div} b \in L^{1}([0,T]; L^{\infty}(\mathbb{R^{N}}))$
hold for the linear transport equation:
$\frac{\partial u}{\partial t} - b \cdot \nabla_{x}u + c u = 0$ in $(0,T) \times \mathbb{R^{N}}$ ;
with the initial condition $u_{0}(x) = u(0,x) \in L^{p}(\mathbb {R^{N}})$.
Now, with this in hand how can we show,
$a)$ $||u(t)||_{\infty} \leq \mbox{C}_{0}||u_{0}||_{\infty}$ a.e. on $(0,T)$, where $\mbox{C}_{0}$ depends only on $||c||_{L^{1}([0,T];L^{\infty}(\mathbb R^{N}))}$
and
$b)$ $||u(t)||_{p} \leq \mbox{C}_{0}||u_{0}||_{p}$ a.e. on $(0,T)$ for $p < \infty$, where $\mbox{C}_{0}$ depends only on $||c + \frac{1}{p} \mbox{div} b||_{L^{1}([0,T];L^{\infty}(\mathbb R^{N}))}$
Now, I know in the case of Linear Transport Equation with constant coefficient (i.e. instead of $b(t,x)$, we would have $b \in \mathbb R^{N}$ only) we have explicitly:
$u(t,x) = u(0,x-tb) + \int_{0}^{t} c(x+(s-t)b,s) ds$ as the solution, from where the estimates follow.
But how to do it in the case with coefficients $b(t,x)$ instead of just $b$ ??
Thanking you,