3.4 Abstract evolution operators

A general framework for treating evolution problems is based on methods from functional analysis. Here, one considers a linear operator A : D (A ) ⊂ X → X with dense domain, ------ D (A ) = X, in a Banach space X and asks under which conditions the Cauchy problem
ut(t) = Au (t), t ≥ 0, (3.125 ) u(0) = f, (3.126 )
possesses a unique solution curve, i.e., a continuously differentiable map u : [0,∞ ) → D (A ) ⊂ X satisfying Eqs. (3.125View Equation, 3.126View Equation) for each f ∈ D (A). Under a mild assumption on A this turns out to be the case if and only if10 the operator A is the infinitesimal generator of a strongly continuous semigroup P (t), that is, a map P : [0,∞ ) → ℒ (X ), with ℒ (X ) denoting the space of bounded, linear operators on X, with the properties that
  1. P (0 ) = I,
  2. P (t + s) = P (t)P (s) for all t,s ≥ 0,
  3. lim P (t)u = u t→0 for all u ∈ X,
  4. { 1 } D (A) = u ∈ X : lit→m0 t [P (t)u − u] exists in X and 1 Au = lit→m0 t [P (t)u − u], u ∈ D (A).

In this case, the solution curve u of the Cauchy problem (3.125View Equation, 3.126View Equation) is given by u (t) = P (t)f, t ≥ 0, f ∈ D (A ). One can show [327Jump To The Next Citation Point, 51Jump To The Next Citation Point] that P (t) always possesses constants K ≥ 1 and α ∈ ℝ such that

αt ∥P (t)∥ ≤ Ke , t ≥ 0, (3.127 )
which implies that ∥u(t)∥ ≤ Ke αt∥f∥ for all f ∈ D (A ) and all t ≥ 0. Therefore, the semigroup P(t) gives existence, uniqueness and continuous dependence on the initial data.

There are several results giving necessary and sufficient conditions for the linear operator A to generate a strongly continuous semigroup; see, for instance, [327Jump To The Next Citation Point, 51Jump To The Next Citation Point]. One useful result, which we formulate for Hilbert spaces, is the following:

Theorem 4 (Lumer–Phillips). Let X be a complex Hilbert space with scalar product (⋅,⋅), and let A : D (A) ⊂ X → X be a linear operator. Let α ∈ ℝ. Then, the following statements are equivalent:

  1. A is the infinitesimal generator of a strongly continuous semigroup P (t) such that ∥P (t)∥ ≤ e αt for all t ≥ 0.
  2. A − αI is dissipative, that is, Re (u,Au − αu ) ≤ 0 for all u ∈ D (A ), and the range of A − λI is equal X for some λ > α.

Example 24. As a simple example consider the Hilbert space X = L2 (ℝn) with the linear operator A : D (A) ⊂ X → X defined by

2 2 n D (A) := {u ∈ X : (1 + |k |)ℱ u ∈ L (ℝ )}, Au := Δu = − ℱ − 1(|k|2ℱ u), u ∈ D (A ),
where ℱ denotes the Fourier–Plancharel operator; see Section 2. Using Parseval’s identity, we find
Re (u,Au ) = Re(ℱ u,− |k|2ℱ u ) = − ∥(|k|ℱ u)∥2 ≤ 0, (3.128 )
hence A is dissipative. Furthermore, let 2 n v ∈ L (ℝ ), then
( ℱ v ) u := ℱ− 1 -------- (3.129 ) 1 + |k |2
defines an element in D (A) satisfying (I − A )u = u − Δu = v. Therefore, the range of A − I is equal to X, and Theorem 4 implies that A = Δ generates a strongly continuous semigroup P (t) on X such that ∥P (t)∥ ≤ 1 for all t ≥ 0. The curves u(t) := P (t)f, t ≥ 0, f ∈ L2(ℝn ) are the weak solutions to the heat equation on ℝn; see Section 3.1.2.

In general, the requirement for A − αI to be dissipative is equivalent to finding an energy estimate for the squared norm E := ∥u∥2 of u. Indeed, setting u(t) := P (t)f and using ut = AP (t)f we find

d d 2 2 --E (t) = --∥u(t)∥ = 2Re (u(t),Au (t)) ≤ 2α∥u (t)∥ = 2 αE (t) (3.130 ) dt dt
for all t ≥ 0 and f ∈ D (A ), which yields the estimate
∥u(t)∥ ≤ eαt∥f ∥, t ≥ 0, (3.131 )
for all f ∈ D (A). Given the dissipativity of A − αI, the second requirement, that the range of A − λI is X for some λ > α, is equivalent to demanding that the linear operator A − λI : D (A ) → X be invertible. Therefore, proving this condition requires solving the linear equation
Au − λu = v (3.132 )
for given v ∈ X. This condition is important for the existence of solutions, and shows that for general evolution problems, requiring an energy estimate is not sufficient. This statement is rather obvious, because given that A − αI is dissipative on D (A ), one could just make D (A ) smaller, and still have an energy estimate. However, if D (A ) is too small, the Cauchy problem is over-determined and a solution might not exist. We will encounter explicit examples of this phenomenon in Section 5, when discussing boundary conditions.

Finding the correct domain D (A ) for the infinitesimal generator A is not always a trivial task, especially for equations involving singular coefficients. Fortunately, there are weaker versions of the Lumer–Phillips theorem, which only require checking conditions on a subspace D ⊂ D (A ), which is dense in X. It is also possible to formulate the Lumer–Phillips theorem on Banach spaces. See [327, 152, 51Jump To The Next Citation Point] for more details.

The semigroup theory can be generalized to time-dependent operators A (t), and to quasilinear equations where A(u ) depends on the solution u itself. We refer the reader to [51] for these generalizations and for applications to examples from mathematical physics including general relativity. The theory of strongly continuous semigroups has also been used for formulating well-posed initial-boundary value formulations for the Maxwell equations [354Jump To The Next Citation Point] and the linearized Einstein equations [309Jump To The Next Citation Point] with elliptic gauge conditions.

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