dissipative operator meaning

Question

Can someone explain to me the "meaning" of the dissipative operator ?

https://en.wikipedia.org/wiki/Dissipative_operator

I am a bit confused. Thanks in advance.

Answer 1

When we deal with some systems of PDEs (e.g., hyperbolic systems) via semigroup theory, we rewrite the system as an abstract Cauchy problem $$\left\{\begin{align}&\frac{d}{dt}U=AU\\ &U(0)=U_0\end{align}\right.$$ where $A$ is an operator defined in a subspace of a suitable Hilbert space $(H,\|\cdot\|_H)$. In this abstract setting, the space $H$ is chosen so that the energy of the system satisfies $$E(t)=\|U(t)\|_H^2=\langle U(t),U(t)\rangle_H.$$ As a consequence, $$\frac{d}{dt}E(t)=\left\langle \frac{d}{dt}U(t),U(t)\right\rangle_H+\left\langle U(t),\frac{d}{dt}U(t)\right\rangle_H=2\text{Re}\Big(\langle AU(t),U(t)\rangle_H\Big).$$ So, if $A$ is dissipative (that is, if it satisfies $\text{Re}(\langle Ax,x\rangle_H)\leq 0$ for all $x$) then the energy of the system has nonpositive derivative. Thus, in this context, an operator be dissipative means that the energy of the system is nonincreasing.

For other interpretations (and references on the interpretation above), I suggest that you take a look at the section 7 (pages 30-39) of the paper Recent developments in some non-self-adjoint problems of mathematical physics by C. L. Dolph.