Strongly continuous semigroups and symbols of pseudo differential operators

Question

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 :)

Answer 1

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.