- A strongly continuous semigroup (or \(C^0\)-semigroup) satisfies
\begin{align*}
T(t+s) \amp=\amp T(t) T(s), \qquad s,t\geq 0,\\
T(0) \amp=\amp I,\\
t\amp\mapsto \amp T(t) \quad \mbox{ is continuous at }0.
\end{align*}
- If \(\{T(t)\}\) is a strongly continuous semigroup, then
\begin{equation*}
\|T(t) \|\leq M e^{\omega t}.
\end{equation*}
- The infinitesimal generator \(A\) is defined as
\begin{equation*}
Ax := \lim_{h\to 0^+} \frac{T(h) x- x}{h} .
\end{equation*}
It satisfies
\begin{equation*}
\frac{d}{dt} T(t) = A \,T(t)x, \qquad T(0)=x_0.
\end{equation*}
- The inhomogeneous problem
\begin{equation*}
\dot u = Au + f(t) , \quad u(0)=u_0
\end{equation*}
has mild solutions that are given by
\begin{equation*}
u(t) = T(t) u_0 +\int_0^t T(t-s) f(s) ds.
\end{equation*}
- \(A\)
- \(A\) is closed and \(D(A)\) is dense.
- For all \(\lambda > \omega\) we have a resolvent estimate
\begin{equation*}
\|R_\lambda(A)^n \| \leq \frac{M}{(\lambda-\omega)^n}.
\end{equation*}
\begin{equation*}
T(t) = \lim_{n\to \infty} \left(I-\frac{t}{n} A\right)^{-n}= e^{At}
\end{equation*}
\begin{equation*}
R_\lambda(A) x = -\int_0^\infty e^{-\lambda t} T(t) x dt.
\end{equation*}
- If \(M=1\) in Hille-Yosida, then we only need
\begin{equation*}
\|R_\lambda(A)\|\leq \frac{1}{\lambda-\omega} \quad \mbox{ for all } \lambda>\omega.
\end{equation*}
- \(A\)
\begin{equation*}
\|T(t)\|\leq e^{\omega t},
\end{equation*}
- \(A\) is closed and densely defined
- \(\displaystyle \mbox{Re}(x, Ax) \leq \omega(x,x)\)
- There exists a \(\lambda_0 >\omega\) in the resolvent set.
- {\bf Perturbations:} \(A\) generator, \(B\) bounded, then \(A+B\) is a generator with
\begin{equation*}
\|S(t) \|\leq M e^{(\omega +M\|B\|) t}.
\end{equation*}
- \(\Sigma_{\frac{\pi}{2}+\delta}\)
- Analytic semigroups have sectorial generators and sectorial generators generate analytic semigroups.
- The semigroup can be written with a Cauchy integral formula
\begin{equation*}
T(z) = \frac{i}{2\pi} \ointop_\gamma e^{\mu z} R_\mu(A) d\mu,
\end{equation*}
thereby completing the Semigroup Triangle Figure [cross-reference to target(s) "semigrouptriangle3" missing or not unique].
- Analytic semigroups satisfy
\begin{equation*}
\|A e^{At}\|\leq c\frac{e^{\omega t}}{t}.
\end{equation*}
- Analytic semigroups regularize.
- Perturbations with operators that are dominated by \(A\text{.}\)
