Questions: Semigroup Theory for Evolution Equations
4 questions to test your understanding
Score: 0 / 4
Question 1 Multiple Choice
A strongly continuous (C₀) semigroup S(t) on a Banach space X satisfies:
AS(0) = I, S(t+s) = S(t)S(s), and S(t)x → x as t → 0⁺ for all x ∈ X
BS(t)S(s) = S(ts) for all t,s ≥ 0
CS(t) is compact for all t > 0
D||S(t)|| = 1 for all t ≥ 0
These three properties define a C₀ semigroup. The semigroup property S(t+s) = S(t)S(s) expresses the time-translation invariance of autonomous systems. Strong continuity (S(t)x → x pointwise) is the minimal continuity requirement, weaker than operator-norm continuity.
Question 2 True / False
The Hille-Yosida theorem is the analogue of the Picard-Lindelof existence theorem for ODEs in infinite-dimensional spaces.
TTrue
FFalse
Answer: True
Just as Picard-Lindelof guarantees existence and uniqueness for u' = f(t,u) with Lipschitz f, the Hille-Yosida theorem guarantees existence of a semigroup (= solution operator) for u' = Au when A satisfies appropriate resolvent estimates. It is the foundational existence result for linear evolution equations.
Question 3 Short Answer
What is the infinitesimal generator of the heat semigroup on L²(Ω)?
Think about your answer, then reveal below.
Model answer: The Laplacian Δ with appropriate domain (D(A) = H²(Ω) ∩ H¹₀(Ω) for Dirichlet conditions)
The heat semigroup S(t)u₀ = e^{tΔ}u₀ solves the heat equation. Its generator A is defined by Au = lim_{t→0⁺}(S(t)u - u)/t = Δu, with domain the set of u ∈ L² where this limit exists, which is H² ∩ H¹₀ by elliptic regularity.
Question 4 True / False
Semigroup theory can only handle linear evolution equations.
TTrue
FFalse
Answer: False
While the basic theory is for linear semigroups, nonlinear evolution equations u_t = Au + F(u) are handled by treating F as a perturbation and using fixed-point arguments (mild solutions) or the theory of nonlinear semigroups. The Crandall-Liggett theorem generates nonlinear semigroups from accretive operators.