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--- hamburgminicourse2017:the_abstract_cauchy_problem_in_banach_space [2017/04/02 13:59] – markus
+++ hamburgminicourse2017:the_abstract_cauchy_problem_in_banach_space [2017/04/02 15:57] (current) – markus
@@ Line 18: / Line 18: @@
 with domain $D(A)$, i.e. all $x\in X$ for which the limit exists.
-Theorems that show the existence of such a semigroup by demanding specific properties from its generator are called //generation theorems//. An example is Stone's theorem in the Hilbert space case and the theorems of Hille-Yoshida and Lumer-Phillips in the more general setting of Banach spaces. The important relation between the semigroup and the Cauchy problem above is given by the following theorem with two different types of solutions (discussed later).
+Theorems that show the existence of such a semigroup by demanding specific properties from its generator are called //generation theorems//. An example is [[Stone's theorem]] in the Hilbert space case and the theorems of Hille--Yoshida and Lumer--Phillips in the more general setting of Banach spaces. The important relation between the semigroup and the Cauchy problem above is given by the following theorem with two different types of solutions (discussed later).
 **Theorem.** Let $T(t)$ be a $\mathcal{C}^0$ semigroup and $A$ its infinitesimal generator. Then it holds for $t>0$ that
@@ Line 52: / Line 52: @@
 **Proof.**  Like before it holds
 \[
-x = \lim_{h \searrow 0}\frac{1}{h} \int_0^h T(s) x \d s
+x = \lim_{h \searrow 0}\frac{1}{h} \int_0^h T(s) x \,ds
 \]
 and by (1.) above the argument of the limit is in $D(A)$. Hence any $x \in X$ can be approached by a limit sequence and $D(A)$ is dense in $X$. Assume now such a sequence $x_n \in D(A), x_n \rightarrow x$ and $Ax_n \rightarrow y$. Then integrating out part (2.) over the time interval $[0,h]$ we can write
@@ Line 67: / Line 67: @@
 \]
 This shows closedness. $\Box$
+We can find two **types of solutions** to the Cauchy problem formulated here:
+  - $u \in (0,\infty) \rightarrow D(A)$ continuous in $X$ and continuously differentiable that really solve $\frac{d}{dt}u(t)=Au(t)$ with $\lim_{t\rightarrow 0}u(t)=x\in X$ are called //classical// or //strong solutions//.
+  - $u(t) = T(t)x$ (also continuous in $X$) just defined by application of the corresponding evolution semigroup are called //generalised solutions// and can be defined for any initial state $x\in X$. They solve a weaker form of the Cauchy problem from (2.) in the theorem above:
+\[
+A\int_0^t u(s)\,ds = u(t)-x.
+\]
+(Such solutions of an integral equation are called //mild//.)
+ >> [[Stone's theorem]]