Let $p\in [1,\infty)$. On $X=L^p(\mathbb{R}; \mathbb{C}^2)$ we consider the operator $D=A+B$, where
$A=\begin{pmatrix}-\partial_x & 0 \\ 0 & \partial_x \end{pmatrix}\quad{and}\quad B=c(x)\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$
(with domain $\mathsf{Dom}(D)=W^{1,p}(\mathbb{R};\mathbb{C}^2)$). Here, $c\colon \mathbb{R}\to \mathbb{R}$ is a real-valued $L^\infty$-function. My question is:
Does $D$ generate a bounded $C_0$-group on $X$?
My thoughts so far:
It is clear that $A$ generates a bounded $C_0$-group on $X$. In fact,
$(e^{tA}f)(x)=\begin{pmatrix} f_1(x-t)\\ f_2(x+t)\end{pmatrix}$ for a.e. $x\in \mathbb{R}$, all $t\in \mathbb{R}$ and all $f=(f_1,f_2)\in X$,
from which we immediately get that $(e^{tA})_{t\in \mathbb{R}}$ is a $C_0$-group of contractions on $X$.
Now, as $B$ is bounded linear operator on $X$, it follows from standard pertubation theory that $D$ is the generator of a $C_0$-group $(e^{tD})_{t\in \mathbb{R}}$. However, pertubation theory only guarantees a growth rate $\omega\geq 0$ and and $M\geq 1$ such that $\|e^{tD}\|\leq Me^{\omega |t|}$ for all $t\in \mathbb{R}$ and this $\omega$ can be strictly positive in general (consider for example the case where $B$ was the identity matrix). So, the question is if $\omega$ can be chosen to be zero, and this seems to be a much more delicate matter.
If $f=(f_1,f_2)\in \mathsf{Dom}(D)$, then $(u,v)(t):=e^{tD}f$ for $t\in \mathbb{R}$ solves the abstract Cauchy problem
$(\ast) \begin{cases} u'(t)=-\partial_x u(t)+cv(t) \quad (t\in \mathbb{R}), \quad u(0)=f_1,\\ v'(t)=\hphantom{-}\partial_x u(t)-cu(t) \quad (t\in \mathbb{R}), \quad v(0)=f_2. \end{cases} $
The boundedness of $(e^{tD})_{t\in \mathbb{R}}$ is then equivalent to the existence of $M\geq 1$ such that the estimate
$\|(u,v)(t)\|_{X}\leq M \|(f_1,f_2)\|_X$ for all $t\in \mathbb{R}$
holds. The problem is that I am not able to find any reasonable representation of the solution of $(\ast)$ which allows me estimate its $X$-norm in an effective way. Here, the problem is that $(\ast)$ is coupled (it is a coupled system of transport equations) and that $c$ is only asssumed to be $L^\infty$ which makes taking one additional time derivative in order to obtain a decoupled wave equation problematic.
My hope is that $(e^{tD})_{t\in \mathbb{R}}$ is indeed bounded, and if not I am curious to know if it is possible to give (smallness-)conditions on $c$ that make the $C_0$-group bounded.
Any hints would be greatly appreciated.