I was looking for references on the following PDEs: $$ u_t + a u_x - w_{xx} = f \\ w_t + a w_x -u =g \\ u(x,0)=u_0(x), \ w(x,0)=w_0(x) $$ where $f\in C([0,T];H^s(\mathbb{R}))$; $a,g\in C([0,T];H^{s+1}(\mathbb{R}))$; $u_0\in H^s(\mathbb{R})$; $w_0\in H^{s+1}(\mathbb{R})$. Both $u$ and $w$ are scalar functions. The index $s\geq 3$.
I was wondering how should we prove the existence and uniqueness of solutions $u\in C([0,T];H^{s}(\mathbb{R}))$ and $w\in C([0,T];H^{s+1}(\mathbb{R}))$?
I'd appreciate it a lot for any advice! Thank you very much!
\begin{align} v_{xt} + a v_{xx} - w_{xx} = f \implies v_{t} + a v_{x} - w_{x} &= F \ w_{t} + a w_{x} - v_{x} &= g \end{align}
with $F' = f$, which can then be solved using the method of characteristics, either by diagonalising the system or by adding the two equations together directly and defining $z = v + w$.
– Matthew Cassell Jul 05 '23 at 11:01