Let $x \in (0,L)$, $t \in (0,T)$, and let $P = P(x,t) \in \mathbb{R}^{3\times 3}$, $Q = Q(x,t) \in \mathbb{R}^{3\times 3}$, $R_0 = R_0(x) \in \mathbb{R}^{3\times 3}$ and $G= G(t) \in \mathbb{R}^{3 \times 3}$ be continuous functions.
My question is:
Can we find a function $R = R(x,t) \in \mathbb{R}^{3 \times 3}$ that satisfies \begin{equation} \begin{cases} \partial_t R(x,t) = R(x,t) P(x,t) & \text{in } (0,L)\times(0,T)\\ \partial_x R(x,t) = R(x,t) Q(x,t) & \text{in } (0,L)\times(0,T)\\ R(x,0) = R_0(x) & \text{for }x \in (0,L)\\ R(0,t) = G(t) & \text{for }t \in (0,T). \end{cases} \end{equation} with the supplementary assumptions: \begin{align*} & 1. \text{ $R$ is unitary (i.e. $RR^\intercal = R^\intercal R = I$, where $I$ is the identity matrix),}\\ & 2. \text{ the determinant of $R$ is equal to one,}\\ & 3. \text{ $P$ and $Q$ are both skew-symmetric (i.e. $Q^\intercal = -Q$ and $P^\intercal = -P$).}\\ \end{align*}
(Here $\partial_t$ and $\partial_x$ denote the partial derivative with respect to time and space respectively.)
Following ideas from the scalar case (see Two PDE for one unknown?), I first considered, for $x \in (0, L)$ fixed, the solution to the initial value problem $\partial_t R = R P$: \begin{align*} R(x,t) = R_0(x) \exp\left( \int_0^t P(x,s)ds \right), \end{align*} and for $t \in (0,T)$ fixed, the solution to the initial value problem $\partial_x R = R Q$: \begin{align*} R(x,t) = G(t) \exp \left( \int_0^x Q(y,t) dy \right). \end{align*} I was looking for conditions on the coefficients $P$, $Q$ and on the initial and boundary data, sufficient to imply the existence of a solution $R$ (e.g. $G'(t) = G(t) Q(0,t)$, and others). However, the fact that matrices (and not scalars) are involved implies that terms do not necessarily commute (e.g. $R$ and $\partial_t R$ do not necessarily commute).
Any suggestion of method, reference, explanation of why it is/is not possible to have a solution, would be welcome. Thank you.
Edit: I realize now that the formula I gave (above) for the solutions of the initial value problems are probably not correct without assuming that some quantities involving $P(x,t)$ and $Q(x,t)$ commute.
