I am trying to better understand the relationships between classical solutions for PDEs and for abstract differential equations. In particular, let us consider a very simple case: the transport equation $$ u_t=u_x\qquad \forall (x,t)\in (0,1)\times\mathbb{R}_{>0} $$ Let $u^\star\colon\mathcal{C}^1((0,1)\times\mathbb{R}_{>0};\mathbb{R})$ be a solution to the above PDE. Now, we know that the above PDE can be rewritten as the following abstract differential equation $$ \dot{u}=Au $$ where $A\colon\mathcal{D}(A)\rightarrow\mathcal{C}^0((0,1);\mathbb{R})$ is the standard differential operator. Now, here is my question: Let $t\mapsto v(t)=u^\star(t,\cdot)$, where $u^\star$ is the solution to the above transport PDE, can one claim that for all $t\in\mathbb{R}_{>0}$, $\dot{v}(t)=Av(t)$?
Thanks!!