Let $d$ be some given integer, I am considering the following system of equations
\begin{align*} \begin{cases} \partial_t v_i(t)=b_i(t,v(t))-\sum_{j=1}^d\partial_{x_j}v_j,\; (t,x)\in[0,T]\times \mathbb R^d\\ v_i(0,x)=\varphi_i(x),\;\;for\; i=1,...,d. \end{cases} \end{align*}
which can be written as
\begin{align*} \begin{cases} \partial_t v(t)=b(t,v(t))-\sum_{j=1}^d\mathcal I_j\:\partial_{x_j}v,\;(t,x)\in[0,T]\times \mathbb R^d\\ v(0,x)=\varphi(x) \end{cases} \end{align*}
where $\mathcal I_j$ is a matrix in which the $j$-th column is full of $1$s and zeroes elsewhere. The problem is that this system is not symmetric nor symmetrizable and hence the theory proposed by Friedrichs, Lax, Majda, etc. does not apply.
Are you aware of some way to deal with this kind of equations? Is there a theory for non-symmetric system of PDEs of the first order?
Thanks in advance!
EDIT: I tried to delete my answer because it is wrong: See Courant and Hilbert, Methods of Mathematical Physics Vol II, page 140. Your equations have the "same principal part," and are equivalent as shown there to a homogeneous linear differential equation for a function of $2d$ variables ($x_1,\ldots,x_d$ and $v_1,\ldots,v_d$), and to a system of $2d$ ordinary differential equations.