I am considering the Cauchy IVP for the evolution equation
$$u\_t + \Psi u =0$$
where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. 


The corresponding solution semigroup is
$$u = \mathcal{F}^{-1}\left( e^{-\widehat{\Psi} t} \mathcal{F}(f) \right)$$


What are the conditions under which a linear pseudo-differential operator generates a strongly continuous semigroup in terms of its symbol? I was thinking there would be a way in which the Hille-Yosida conditions could be rewritten in terms of symbols rather than operators.


Thanks :)



Strongly continuous semigroups and symbols of pseudo differential operators
The answer may depend upon which functional space you are dealing with. But since you insist upon the symbol and the Fourier transform, let me assume that you have $L^2({\mathbb R}^d)$ in mind. Because $\cal F$ is an isometry, it is equivalent to ask whether $t\mapsto[\xi\mapsto\exp(-t\hat\Psi(\xi))]$ is a strongly continuous semigroup. The answer is simply that the real part of $\hat\Psi(\xi)$ be bounded by below as $\xi$ runs over ${\mathbb R}^d$. The necessity is obvious. The only delicate point in sufficiency is the continuity
$$\lim\_{t\rightarrow0+}\|\exp(-t\hat\Psi(\cdot))f-f\|\_{L^2}=0,\qquad\forall f\in L^2.$$
This is proved by dominated convergence.


The answer is exactly the same in Sobolev spaces $H^s({\mathbb R}^d)$.


On the contrary, many Cauchy problems that are well-posed in $L^2$ turn out to be ill-posed in $L^p$ for $p\ne2$. This is the case of the wave equation, which can be recast as a first-order equation with symbol $i|\xi|$. For conservative equations (those for which the symbol is pure imaginary), this is related to Strichartz inequalities. See the first chapter of the book that I co-authored with S. Benzoni-Gavage.


