Strongly Continuous Semigroups — Mathematics Knowledge System

Overview

A strongly continuous (C_0) semigroup on a Banach space X is a family of bounded operators {T(t)}_{t >= 0} satisfying T(0) = I, T(s+t) = T(s)T(t), and lim_{t->0+} T(t)x = x for each x in X. C_0 semigroups provide the abstract framework for solving linear evolution equations du/dt = Au where A is an unbounded operator (like a differential operator). The Hille–Yosida theorem characterises which operators A generate a C_0 semigroup.

Intuition

A C_0 semigroup is the infinite-dimensional version of the matrix exponential e^{tA}. For an ODE du/dt = Au with A a matrix, the solution is u(t) = e^{tA}u(0). For a PDE like the heat equation u_t = Delta u, we want u(t) = T(t)u(0) where T(t) is the 'heat flow' operator. The semigroup property T(s+t) = T(s)T(t) encodes the time-translation invariance, and strong continuity says the solution depends continuously on time.

Formal Definition

Definition

A family {T(t)}_{t >= 0} in B(X) is a C_0 semigroup if: (1) T(0) = I; (2) T(s+t) = T(s)T(t) for all s,t >= 0; (3) lim_{t->0+} ||T(t)x - x|| = 0 for all x in X. The infinitesimal generator A of the semigroup is the closed unbounded operator defined by Ax = lim_{t->0+} (T(t)x - x)/t on its domain D(A) = {x in X : the limit exists}. The solution to du/dt = Au, u(0) = x_0 is u(t) = T(t)x_0 for x_0 in D(A).

T(0) = I,\quad T(s+t) = T(s)T(t),\quad \lim_{t\to 0^+}\|T(t)x - x\| = 0

C_0 semigroup axioms

Ax = \lim_{t\to 0^+}\frac{T(t)x - x}{t},\quad D(A) = \left\{x\in X : \text{limit exists}\right\}

Infinitesimal generator

\|T(t)\| \le M e^{\omega t}\quad \text{for some } M\ge 1,\, \omega \in \mathbb{R}

Exponential bound

Notation

Notation	Meaning
$\{T(t)\}_{t\ge 0}$	C_0 semigroup
$A$	Infinitesimal generator of the semigroup
$D(A)$	Domain of the generator A
$R(\lambda, A)$	Resolvent of A at lambda

Theorems

Theorem 10.1: Theorem 10.1 (Hille–Yosida)

A closed densely defined operator A on a Banach space X generates a C_0 semigroup satisfying ||T(t)|| <= M e^{omega t} iff the resolvent set of A contains (omega, infinity) and ||R(lambda, A)^n|| <= M/(lambda - omega)^n for all lambda > omega and n >= 1.

Theorem 10.2: Theorem 10.2 (Lumer–Phillips)

A densely defined operator A generates a contraction semigroup (||T(t)|| <= 1) iff A is dissipative (Re<Ax,x*> <= 0 for all x in D(A), x* in J(x) the duality map) and lambda I - A is surjective for some lambda > 0.

Theorem 10.3: Theorem 10.3

If A generates {T(t)}, then for x in D(A): d/dt T(t)x = AT(t)x = T(t)Ax. The map t |-> T(t)x is the unique classical solution to du/dt = Au, u(0) = x.

Worked Examples

1
T(0)f = f: (T(0)f)(x) = f(x+0) = f(x). Check.
2
Semigroup property: (T(s)T(t)f)(x) = (T(t)f)(x+s) = f(x+s+t) = (T(s+t)f)(x). Check.
3
Strong continuity: ||T(t)f - f||_p^p = integral |f(x+t) - f(x)|^p dx -> 0 as t -> 0 for f in L^p(R).
$\|T(t)f - f\|_p \to 0 \text{ as } t \to 0$
4
This follows because continuous functions with compact support are dense in L^p and translation is continuous for such functions, then use a density argument.
5
The generator A is d/dx (differentiation), with domain {f in L^p : f' in L^p}.

✓ Answer

The translation semigroup T(t)f(x) = f(x+t) is a C_0 contraction semigroup on L^p(R) with generator d/dx.

Practice Problems

Mediumproof writing

Show that the generator A of a C_0 semigroup {T(t)} is a closed operator.

Hardfree response

State the Hille–Yosida theorem and explain how it is used to show the heat equation u_t = Delta u on L^2(R^n) is well-posed.

Quiz

Summary

A C_0 semigroup satisfies T(0)=I, T(s+t)=T(s)T(t), and strong continuity at t=0.
The generator A = lim_{t->0+}(T(t)x - x)/t is a closed densely defined (usually unbounded) operator.
C_0 semigroups solve linear evolution equations: u(t) = T(t)u_0 satisfies du/dt = Au.
Hille–Yosida theorem: A generates a bounded C_0 semigroup iff resolvent estimates hold.
Lumer–Phillips: A generates a contraction semigroup iff A is dissipative with full range.

References

BookPazy, A. — Semigroups of Linear Operators and Applications to PDEs, Springer, 1983
BookEngel, K.-J. & Nagel, R. — One-Parameter Semigroups for Linear Evolution Equations, Springer, 2000
Websiteen.wikipedia.org