Semigroups and Evolutionary Equations

We show how strongly continuous semigroups can be associated with evolutionary equations. For doing so, we need to define the space of admissible history functions and initial states. Moreover, the initial value problem has to be formulated within the framework of evolutionary equations, which is done by using the theory of extrapolation spaces. The results are applied to two examples. First, differential-algebraic equations in infinite dimensions are treated and it is shown, how a C_{0}-semigroup can be associated with such problems. In the second example we treat a concrete hyperbolic delay equation.


Introduction
In this article we bring together two theories for dealing with partial differential equations: the theory of C 0 -semigroups on the one hand and the theory of evolutionary equations on the other hand. In particular, we show how C 0 -semigroups can be associated with a given evolutionary equation. The framework of evolutionary equations was introduced in the seminal paper [12]. Evolutionary equations are equations of the form where ∂ t denotes the temporal derivative, M (∂ t ) is a bounded operator in space-time defined via a functional calculus for ∂ t and A is an, in general, unbounded spatial operator. The function F defined on R and taking values in some Hilbert space is a given source term and one seeks for a solution U of the above equation. Here, the notion of solution is quite weak, since one just requires that the solution should belong to some exponentially weighted L 2 -space. Thus, all operators have to be introduced in these spaces. Especially, the time derivative is introduced as an unbounded normal operator on such a space and so, in order to solve (1), one has to deal with the sum of two unbounded operators (∂ t and A). Problems of the form (1) cover a broad spectrum of different types of differential equations, such as hyperbolic, parabolic, elliptic and mixed-type problems, integro-differential equations [22], delay equations [8] and fractional differential equations [14]. Also, generalisations to some nonlinear [18,19] and non-autonomous problems [15,28,29,24] are possible. The solution theory is quite easy and just relies on pure Hilbert space theory.
On the other hand, there is the well-established theory of C 0 -semigroups dealing with so-called Cauchy problems (see e.g. [7,11,5]). These are abstract equations of the form where A is a suitable operator acting on some Banach space. Although, (2) just seems to be a special case of (1) for M (∂ t ) = 1, the theories are quite different. While we focus on solutions lying in L 2 in the theory of evolutionary equations, one seeks for continuous solutions in the framework of C 0 -semigroups. Moreover, while (1) holds on R as time horizon, (2) just holds on R ≥0 and is completed by an initial condition. Thus, in order to associate a C 0 -semigroup with equations of the form (1) one has to find a way to formulate initial value problems and then derive assumptions, which would yield the additional regularity for the solutions (namely continuity with respect to time). This is the purpose of this work. As we have indicated above, equations of the form (1) also cover delay equations, where it is more natural to prescribe histories instead of an initial state at time 0. Moreover, (1) also covers so-called differential algebraic equations (see [9] for the finite-dimensional case and [27,26,25] for infinite dimensions), where not every element of the underlying state space can be used as an initial state. Thus, one is confronted with the problem of defining the 'right' initial values and histories for (1) depending on the operators involved. Moreover, one has to incorporate these initial conditions within the framework of evolutionary equations, that is, initial conditions should enter the equation as a suitable source term on the right-hand side. This can be done by using extrapolation spaces and by extending the solution theory to those. Then it will turn out that initial conditions can be formulated by distributional right hand sides, which belong to a suitable extrapolation space associated with the time derivative operator ∂ t . Having the right formulation of initial value problems at hand, one can associate a C 0 -semigroup on a product space consisting of the current state in the first and the past of the unknown in the second component. This idea was already used to deal with delay equations within the theory of C 0 -semigroups, see [3]. As it turns out, this product space is not closed (as a subspace of a suitable Hilbert space) and in order to extend the associated C 0 -semigroup to its closure one needs to impose similar conditions as in the Hille-Yosida Theorem. The key result, which will be used to extend the semigroup is the theorem of Widder-Arendt (see [1] or Theorem 6.4 below). The paper is structured as follows: We begin by recalling the basic notions and wellposedness results for evolutionary problems (Section 2) and for extrapolation spaces (Section 3). Then, in order to formulate initial value problems within the framework of evolutionary equations, we introduce a cut-off operator as an unbounded operator on the extrapolation space associated with the time derivative and discus some of its properties (Section 4). Section 5 is then devoted to determine the 'right' space of admissible histories and initial values for a given evolutionary problem. We note here that we restrict ourselves to homogeneous problems in the sense that we do not involve an additional source term besides the given history. The main reason for that is that such source terms would restrict and change the set of admissible histories, a fact which is well-known in the theory of differential-algebraic equations. In Section 6 we associate a C 0 -semigroup on the before introduced product space of admissible initial values and histories and prove the main result of this article (Theorem 6.7). In the last section we discuss two examples. First, we apply the results to abstract differential algebraic equations and thereby re-prove the Theorem of Hille-Yosida as a special case. In the second example, we discuss a concrete hyperbolic delay equation and prove that we can associate a C 0 -semigroup with this problem. Throughout, every Hilbert space is assumed to be complex and the inner product ·, · is conjugate-linear in the first and linear in the second argument.

Evolutionary Problems
We recall the basic notions and results for evolutionary problems, as they were introduced in [12] (see also [13,Chapter 6]). We begin by the definition of the time derivative operator on an exponentially weighted L 2 -space (see also [16]).
Definition. Let ρ ∈ R and H a Hilbert space. We set with the common identification of functions coinciding almost everywhere. Then L 2,ρ (R; H) is a Hilbert space with respect to the inner product Moreover, we define the operator We recall some facts on the operator ∂ t,ρ and refer to [8] for the respective proofs.
Proposition 2.1. Let ρ ∈ R and H a Hilbert space.
(a) The operator ∂ t,ρ is densely defined, closed and linear and C ∞ c (R; H) is a core for ∂ t,ρ .
(b) The spectrum of ∂ t,ρ is given by (c) For ρ = 0 the operator ∂ t,ρ is boundedly invertible with ∂ −1 t,ρ = 1 |ρ| and the inverse is given by (e) The following variant of Sobolev's embedding theorem holds: As a normal operator, ∂ t,ρ possesses a natural functional calculus, which can be described via the so-called Fourier-Laplace transform.
Definition. Let ρ ∈ R and H a Hilbert space. We denote by L ρ the unitary extension of the mapping Remark 2.2. Note that for ρ = 0, the operator L 0 is nothing but the classical Fourier transform, which is unitary due to Plancharel's Theorem (see e.g. [17,Theorem 9.13]).
for t ∈ R, it follows that L ρ is unitary as a composition of unitary operators.
Using the latter proposition, we can define an operator-valued functional calculus for ∂ t,ρ as follows.
Definition. Let ρ ∈ R and H a Hilbert space. Let F : {it + ρ ; t ∈ R} → L(H) be strongly measurable and bounded. Then we define ).
An important class of operator-valued function of ∂ t,ρ are those functions yielding causal operators.
The proof of causality is based on a theorem by Paley and Wiener, which charcterises the functions in L 2 (R ≥0 ; H) in terms of their Laplace transform (see [10] or [17, 19.2 Theorem ]). The independence of ρ is a simple application of Cauchy's Theorem for analytic functions.
(b) It is noteworthy that causal, translation-invariant and bounded operators are always of the form M (∂ t,ρ ) for some analytic and bounded mapping defined on a right half plane (see [6,31]).
Finally, we are in the position to define well-posed evolutionary problems.
Definition. Let ρ 0 ∈ R and H a Hilbert space. Moreover, let M : C Re>ρ 0 → L(H) be analytic and bounded and A : dom(A) ⊆ H → H densely defined, closed and linear. Then we call an equation of the form the evolutionary equation associated with (M, A). The problem is called well-posed if there is ρ 1 > ρ 0 such that zM (z) + A is boundedly invertible for each z ∈ C Re≥ρ 1 and is bounded. Moreover we set s 0 (M, A) as the infimum over all such ρ 1 > ρ 0 .

Extrapolation spaces
In this section we recall the notion of extrapolation spaces associated with a boundedly invertible operator on some Hilbert space H. We refer to [13, Section 2.1] for the proof of the results presented here.
Definition. Let C : dom(C) ⊆ H → H be a densely defined, closed, linear and boundedly invertible operator on some Hilbert space H. We define the Hilbert space equipped with the inner product x, y H 1 (C) := Cx, Cy (x, y ∈ dom(C)).

Moreover, we set
Remark 3.1. Another way to introduce the space H −1 (C) is taking the completion of H with respect to the norm Moreover, the operator possesses a unitary extension, which will again be denoted by C.
Example 3.3. Let ρ = 0 and H a Hilbert space. Then we set Moreover, the Dirac distribution δ t at a point t ∈ R belongs to H −1 ρ (R; C) and for each ϕ ∈ C ∞ c (R; C), which shows the asserted formula. The statement for ρ < 0 follows by the same rationale.
H) is bounded and thus has a unique bounded extension to the whole H −1 ρ (R; H). Proof. The assertion follows immediately by realising that We recall that for a densely defined, closed, linear operator A : dom(A) ⊆ H 0 → H 1 between two Hilbert spaces H 0 and H 1 , the operators A * A and AA * are selfadjoint and positive. Then the moduli of A and A * are defined by is bounded and hence, possesses a bounded extension to H 0 .

Cut-off operators
The main goal of the present section is to extend the cut-off operators χ R ≥t and χ R ≤t for some t ∈ R defined on L 2,ρ (R; H) to the extrapolation space H −1 ρ (R; H). For doing so, we start with the following observation.
Lemma 4.1. Let ρ > 0, t ∈ R and H be a Hilbert space. We define the operators .
Proof. We just prove the formula for The latter representation of the cut-off operators on L 2,ρ (R; H) leads to the following definition on H −1 ρ (R; H).
Definition. Let ρ > 0 and H a Hilbert space. For t ∈ R we define the operators where λ denotes the Lebesgue measure on R. The expression f (t−) is defined analogously.
We conclude this section by some properties of the so introduced cut-off operators. Proposition 4.3. Let H be a Hilbert space, ρ > 0, y ∈ H and s, t ∈ R. Then the following statements hold.
(a) δ s y ∈ dom(P t ) and Here, the support spt f is meant in the sense of distributions.
Proof. (a) We note that ∂ −1 t,ρ δ s y = e 2ρs χ R ≥s y and hence, δ s ∈ dom(P t ). Moreover, (c) Let f ∈ H −1 ρ (R; H) and assume first that spt f ⊆ R ≤t . We first prove that ∂ −1 t,ρ f is constant on R ≥t . For doing so, we define Then V is a closed subspace and for g ∈ L 2,ρ (R; H) we have that and an elementary computation shows and Assume on the other hand that f ∈ ker(P t ) and let ϕ ∈ C ∞ c (R >t ; H). We then compute, using that spt which gives spt f ⊆ R ≤t .

Admissible histories for evolutionary equations
In this section we study evolutionary problems of the following form where M and A are as in Theorem 2.6 and g is a given function on R <0 . The first goal is to rewrite this 'Initial value problem' into a proper evolutionary equations as it is introduced in Section 2. For doing so, we start with some heuristics to motivate the definition which will be made below. In particular, for the moment we will not care about domains of operators.
We will now write (3) as an evolutionary equation for the unknown v := u| R ≥0 , which is the part of u to be determined. For doing so, we first assume that u ∈ H 1 ρ (R; H) for some ρ > 0, which means that v + g ∈ H 1 ρ (R; H). We interpret the first line of (3) as where P 0 is the cut-off operator introduced in Section 4. The latter gives Since v is supported on R ≥0 by assumption and M (∂ t,ρ ) is causal by Proposition 2.4, we infer that M (∂ t,ρ )v is also supported on R ≥0 and so, Hence, we arrive at an evolutionary problem for v of the form Since u = v + g ∈ H 1 ρ (R; H) by assumption, we infer that u is continuous by Proposition 2.1 (e) and hence, the limits v(0+) and g(0−) exist and coincide. Hence, v−χ R ≥0 g(0−) ∈ H 1 ρ (R; H) and vanishes on R <0 . The latter gives where in the last equality we have used that M (∂ t,ρ )(v − χ R ≥0 g(0−)) ∈ H 1 ρ (R; H), hence it is continuous, and vanishes on R ≤0 due to causality. Summarising, we end up with the following problem for v Now, to make sense of (4) we need to ensure that the right hand side is well-defined. In particular, we need that M (∂ t,ρ )χ R ≥0 g(0−) (0+) exists. In order to ensure that, we introduce the following notion.
Definition. Let H be a Hilbert space, ρ 0 ≥ 0 and M : C Re>ρ 0 → L(H) be analytic and bounded. We call M regularising, if for all x ∈ H, ρ > ρ 0 the limit exists. Moreover, for ρ > 0 we define the space As it turns out, this assumption suffices to obtain a well-defined expression on the right hand side of (4).
We are now in the position to define the space of admissible history functions g.

Definition.
Let H be a Hilbert space, ρ 0 ≥ 0 and M : C Re>ρ 0 → L(H) be analytic, bounded and regularising. Moreover, let A : dom(A) ⊆ H → H be densely defined, closed and linear. For notational convenience, we set and Remark 5.2. We have We come back to the heuristic computation at the beginning of this section and show, that for g ∈ His ρ the computation can be made rigorously. and u := v + g. Then spt v ⊆ R ≥0 , u ∈ H 1 ρ (R; H) and satisfies (3).
Proof. Note that by assumption u = v + g ∈ H 1 ρ (R; H) and thus, v = u − g ∈ L 2,ρ (R; H). We prove that spt v ⊆ R ≥0 . For doing so, we compute and hence, spt ∂ −1 t,ρ v ⊆ R ≥0 by causality of S ρ . The latter implies spt v ⊆ R ≥0 . Thus, we have u = g on R <0 and we are left to show For doing so, let ϕ ∈ C ∞ c (R >0 ; dom(A * )). We compute , where in the last line we have used g, A * ϕ = 0, since spt g ⊆ R ≤0 . Moreover, we compute where we have used two times that ϕ(0) = 0. Plugging this formula in the above computation, we infer that which shows the claim.
6 C 0 -semigroups associated with evolutionary problems Throughout this section, let H be a Hilbert space, ρ 0 ≥ 0 and M : C Re>ρ 0 → L(H) analytic, bounded and regularising. Moreover, let A : dom(A) ⊆ H → H be densely defined, closed and linear such that the evolutionary problem associated with (M, A) is well-posed. In this section we aim for a C 0 -semigroup associated with the evolutionary problem for (M, A) acting on a suitable subspace of IV ρ × His ρ for ρ > s 0 (M, A). For doing so, we first need to prove that His ρ is left invariant by the time evolution. The precise statement is as follows.
Proof. We first note that where τ t u := u(t + ·) for u ∈ L 2,ρ (R; H), and hence, The latter gives, employing the causality of M (∂ t,ρ ), The latter yields and Remark 5.2 to derive which yields the desired formula for w. Now h ∈ His ρ follows, since by definition The latter theorem allows for defining a semigroup associated with (M, A).
First we show that T ρ defined above is indeed a strongly continuous semigroup. Then T ρ is a strongly continuous semigroup. More precisely, in H × L 2,ρ (R; H) for each g ∈ His ρ .
Proof. Let g ∈ His ρ and t, s ≥ 0. We set v := S ρ (Γ ρ gδ 0 − K ρ g) and u := v + g. By Theorem 6.1 we have that and thus, Moreover, by the continuity of u and the strong continuity of translation in L 2,ρ .
In the rest of this section we show a characterisation result, when T ρ can be extended to a C 0 -semigroup on the space for some µ ≤ ρ. We first prove a result that is suffices to consider the family T ρ 1 .
In order to extend T ρ 1 to X µ ρ we make use of the Widder-Arendt-Theorem.
Then there is f ∈ L ∞ (R ≥0 ; H) such that f ∞ = M and Remark 6.5. The latter Theorem was first proved by Widder in the scalar-valued case [32] and then generalised by Arendt to the vector-valued case in [1]. It is noteworthy that the latter Theorem is also true in Banach spaces satisfying the Radon-Nikodym property (see [4,Chapter III]) and, in fact, this property of X is equivalent to the validity of Theorem 6.4, see [1,Theorem 1.4].
We now identify the function r mentioned in Theorem 6.4 within the presented framework.
Proposition 6.6. Let ρ > s 0 (M, A) and g ∈ His ρ . We set v := S ρ (Γ ρ gδ 0 − K ρ g) ∈ L 2,ρ (R; H) and Then r g ∈ C ∞ (R >ρ ; H). Moreover, Proof. We note that and hence, the regularity of r g follows. Moreover, where we have used the independence of ρ stated in Theorem 2.6. Hence, for each λ > ρ, where we have used the formula for Γ ρ stated in Remark 5.2.
With these preparations at hand, we can now state and prove the main result of this article.
Theorem 6.7. Let ρ > s 0 (M, A) and T ρ be the semigroup on D ρ associated with (M, A). Moreover, for g ∈ His ρ we set For µ ≤ ρ the following statements are equivalent: (i) T ρ can be extended to a C 0 -semigroup on X µ ρ = D ρ H×L 2,µ (R;H) ⊆ H × L 2,µ (R; H).

Differential-algebraic equations and classical Cauchy problems
In this section we consider initial value problems of the form We assume that the evolutionary problem is well-posed, that is we assume that there is ρ 1 ∈ R ≥0 such that zE + A is boundedly invertible for each z ∈ C Re≥ρ 1 and We again denote the infimum over all such ρ 1 ∈ R ≥0 by s 0 (E, A).
Proof. For x ∈ H, ρ > 0 we have and thus, M is regularising with Γ ρ g = Eg(0−) for each g ∈ H 1 ρ (R ≤0 ; H). Moreover, we have which proves the asserted equalities for His ρ and IV ρ . Finally, let x ∈ IV ρ and g ∈ L 2,µ (R ≤0 ; H) for some µ ≤ ρ with ρ > s 0 (E, A). Then we find a sequence (x n ) n∈N in IV ρ and a sequence (ϕ n ) n∈N in C ∞ c (R <0 ; H) such that x n → x and ϕ n → g in H and L 2,µ (R ≤0 ; H), respectively. Moreover, we set and obtain a sequence (ψ n ) n∈N in H 1 ρ (R ≤0 ; H) with ψ n (0−) = x n for n ∈ N and ψ n → 0 as n → ∞ in L 2,µ (R ≤0 ; H). Consequently, setting g n := ψ n + ϕ n ∈ H 1 ρ (R ≤0 ; H) for n ∈ N we obtain a sequence (x n , g n ) n∈N in D ρ with (x n , g n ) → (x, g) in H × L 2,µ (R; H) and thus, (x, g) ∈ X µ ρ . Since the other inclusion holds obviously, this proves the assertion.
We now inspect the space IV ρ a bit closer. In particular, we are able to determine its closure IV ρ and a suitable dense subset of IV ρ . Then U ⊆ IV ρ and U = IV ρ for each ρ > s 0 (E, A). In particular, IV ρ does not depend on the particular choice of ρ > s 0 (E, A).
Proof. Let ρ > s 0 (E, A), x ∈ U and y ∈ dom(A) with Ax = Ey. Then we compute , which shows hat x ∈ IV ρ by Lemma 7.1. For showing the remaining assertion, we prove that IV ρ ⊆ U . For doing so, let x ∈ IV ρ and set v := S ρ (δEx). Then and since the left-hand side belongs to L 2,ρ (R; H) we infer that v ∈ L 2,ρ (R; dom(A)).
and since v is continuous on R ≥0 and hence, 1 where λ > s 0 (E, A) is fixed. Then y n ∈ U, since and since Ez n → Ez = Ay, we infer that y n → y and hence, y ∈ U .
(ii) There exists M ≥ 1 and ω ≥ ρ such that (iv) The family of functions for t ≥ 0 extends to a C 0 -semigroup on IV ρ .
Proof. We first compute the function r g for g ∈ His ρ as it was defined in Theorem 6.7.
(iii) ⇒ (iv): Since T ρ extends to a C 0 -semigroup on IV ρ × L 2 (R ≤0 ; H) and since we infer that there is M ≥ 1 and ω ∈ R such that and thus, (S ρ (t)) t≥0 extends to a C 0 -semigroup on IV ρ . Moreover, since for each t ≥ 0, x ∈ IV ρ , we obtain the at the end asserted formula .
; H) is continuous and hence, the assertion follows by Proposition 6.3.
Remark 7.4. We remark that in the case of classical Cauchy problems, i.e. E = 1, condition (5) is nothing but the classical Hille-Yosida condition for generators of C 0semigroups (see e.g. [5, Chapter II, Theorem 3.8]). Note that in this case, U = dom(A 2 ) in Proposition 7.2 and hence, IV ρ = U = H.

A hyperbolic delay equation
As a slight generalisation of [3, Example 3.17] we consider a concrete delay equation of the form Here, u attains values in L 2 (Ω) for some open set Ω ⊆ R n as underlying domain, h 0 , . . . , h n > 0 are given real numbers and k, c 0 , . . . , c n are bounded operators on L 2 (Ω) n and L 2 (Ω), respectively. The operators grad and div denote the usual gradient and divergence with respect to the spatial variables and will be introduced rigorously later. It is our first goal to rewrite this equation as a suitable evolutionary problem. For doing so, we need the following definition.
Lemma 7.6. The function M is regularising.
We now rewrite (6) as an evolutionary equation. We introduce v := ∂ t,ρ u and q := k grad u as new unknowns, and rewrite (6) as Of course (6) has to be completed by suitable boundary conditions. This will be done by introducing the differential operators div and grad in a suitable way.
Thus, by replacing div by div 0 or grad by grad 0 in (7), we can model homogeneous Neumann-or Dirichlet conditions, respectively. Proof. The claim follows immediately by the definitions of the differential operators.
We now prove that the evolutionary problems associated with (M, A D/N ) are well-posed.
Proof. We first note that k −1 is selfadjoint and satisfies k −1 ≥ 1 k . Moreover, since A D/N is skew-selfadjoint, we infer that Re A D/N x, x = 0 (x ∈ dom(A D/N )).
Remark 7.10. We note that the above proof also works for m-accretive operators A instead of A D/N . This allows for the treatment of more general boundary conditions and we refer to [20] for a characterisation result about those boundary conditions (including also nonlinear ones).
Having these results at hand, we are now in the position to consider the history space for (7). From now on, to avoid cluttered notation, we will simply write A and note that A can be replaced by A N and A D , respectively. for each µ ≤ ρ.
Thus, we are left to consider the last term. By assumption, we find x ∈ dom(A) with Ag(0−) = 1 0 0 k −1 x and hence, which proves the claim.
We conclude this section by proving that the associated semigroup can be extended to X µ ρ for each µ ≤ ρ.
Proof. The proof will be done by a perturbation argument. For doing so, we consider the evolutionary problem associated with (E, A), where E := 1 0 0 k −1 .
We note that this problem is well-posed with s 0 (E, A) = 0 (compare the proof of Proposition 7.9). We denote the associated semigroup by T ρ . By Proposition 7.2 we know that the closure of the initial value space for T ρ is given by {x ∈ dom(A) ; E −1 Ax ∈ dom(A)} = L 2 (Ω) × L 2 (Ω) n .