Can we convert every PDE into semigroup setting?

Question

Consider the PDE with Cauchy data

$\frac{\partial u}{\partial t}=\frac{\partial u}{\partial x}, ~~ t>0, x \in \mathbb{R}$

$u(0)=v$

Then with $X=L^{2}(\mathbb{R}),$ if we choose $ A \phi = \frac{\partial u}{\partial x} $ with domain $D(A)=H^{1}(\mathbb{R})$, we can convert our PDE into semigroup setting and check the conditions of Hille Yosida Theorem or Lumer Philips Theorem to know whether $A$ will generate a semigroup or not. But I want to know whether we can convert every PDE into semigroup setting or we need some condition on PDE with given initial and boundary data.

Answer 1

If, in your Cauchy setup, on a Banach space $X$, the unbounded linear operator $(A, D(A))$ generates a strongly continuous semigroup (the semigroup $C^0$) then yes you can transform it.