submitted to mediterranean journal of mathematics - …maerceg/documents/papers/burazin... · kre...

Kresimir Burazin∗ & Marko Erceg

Non-stationary abstract Friedrichs systems

Submitted to Mediterranean Journal of Mathematics

Abstract

Inspired by the results of Ern, Guermond and Caplain (2007) on the abstract theoryfor Friedrichs symmetric positive systems, we give the existence and uniqueness resultfor the initial (boundary) value problem for the non-stationary abstract Friedrichs sys-tem. Despite absence of well-posedness result for such systems, there were alreadyattempts for their numerical treatment by Burman, Ern, Fernandez (2010) and Bui-Thanh, Demkowicz, Ghattas (2013). We use the semigroup theory approach, and provethat the operator involved satisfy the conditions of the Hille-Yosida generation theorem.We also address the semilinear problem and apply the new results to a number of ex-amples, such as the symmetric hyperbolic system, the unsteady div-grad problem, andthe wave equation. Special attention was paid to the (generalized) unsteady Maxwellsystem.

Keywords: symmetric positive first-order system, semigroup, abstract Cauchy problem

Mathematics subject classification: 34Gxx, 35F35, 35F40, 35F31, 47A05, 47D06

Department of Mathematics Department of MathematicsUniversity of Osijek University of ZagrebTrg Ljudevita Gaja 6 Bijenicka cesta 30HR31000, Osijek, Croatia HR10000, Zagreb, Croatia+38531224822

[email protected] [email protected]

This work is supported in part by the Croatian Science Foundation trough project Weak Convergence Methodsans Applications, by the J.J. Strossmayer University of Osijek trough project Evolution Friedrichs systems, and byDAAD trough project Centre of Excellence for Applications of Mathematics.

Part of this research was carried out while both of the authors enjoyed the hospitality of the Basque Centrefor Applied Mathematics within the frame of the FP7-246775 NUMERIWAVES project of the European ResearchCouncil.

∗ Corresponding author

21st February 2016

Submitted to Mediterranean Journal of Mathematics Non-stationary abstract Friedrichs systems

1. Introduction

An overview

Symmetric positive systems of first-order linear partial differential equations were introducedby K. O. Friedrichs [21] in 1958 and today they are also known as Friedrichs systems. Hisimmediate goal was the treatment of equations that change their type, like the equations modellingthe transonic fluid flow.

The class of Friedrichs systems encompasses a wide variety of classical and neoclassical linearpartial differential equations of continuum physics, regardless of their type, paired with vari-ous (initial) boundary conditions (Dirichlet, Neumann, Robin). This includes boundary valueproblems for some elliptic systems, mixed initial and boundary value problems for parabolic andhyperbolic equations, and boundary value problems for equations of mixed type, such as theTricomi equation. For some specific examples we refer to [4, 5, 6, 15, 19, 24 27].

The inclusion of such a variety of different problems into a single framework requires alldifferent characteristics to be included as well. This naturally imposes a number of technicaldifficulties, which many authors have since tried to surmount [27, 28].

The setting of symmetric positive systems appeared to be convenient for the numerical treat-ment of various boundary value problems as well. Already Friedrichs considered the numericalsolution of such systems, by a finite difference scheme. The renewed interest in the theory ofFriedrichs systems during the last decade resulted in development of new numerical schemesbased on adaptations of the finite element method. Let us just mention [23], the Ph.D. thesis ofMax Jensen [24] and, most recently, [6, 16, 17, 18]. As we understan, some numerical algorithmsbased on mixed finite element methods are more suitable for first order systems then for higherorder systems, which makes the framework of Friedrichs systems more convenient for numericaltreatment of some higher order equations. By applying some numerical scheme developed forFriedrichs systems to a particular equation of interest, one often gets a numerical result (for thatequation) that was not known before.

Recently, Ern, Guermond and Caplain [16, 19] suggested another approach to the Friedrichssystems, which was, as we understand, inspired by their interest in the numerical treatment ofFriedrichs systems. They expressed the theory in terms of operators acting in abstract Hilbertspaces and represented the boundary conditions in an intrinsic way, thus avoiding the questionof traces for functions from the graph space of the considered operator. Some open questionsregarding the relationship of different representations of boundary conditions in the abstractsetting were answered in [2]. The precise relationship between the classical Friedrichs theory andits abstract counterpart was investigated in [3, 4], which resulted in new applications to secondorder equations, which were also addressed in [5] and [9]. In the first reference, the abstracttheory was applied to the heat equation, while in the second the homogenisation theory of theFriedrichs systems was developed and contrasted to the known results for stationary diffusionequation and the heat equation.

Although some evolution (non-stationary) problems can be treated within the framework ofabstract theory of Friedrichs systems developed in [19], by not making distinction between thetime variable and space variables [5], the theory is not suitable for some standard problems, likethe initial-boundary value problem for non-stationary Maxwell system, or the Cauchy problemfor symmetric hyperbolic systems.

In this paper we develop an abstract theory for non-stationary Friedrichs systems that canaddress these problems as well. More precisely, we consider an abstract Cauchy problem inthe Hilbert space, involving a time independent abstract Friedrichs operator. In order to provethe well-posedness, we use the semigroup theory approach and prove that the operator involvedsatisfies the conditions of the Hille-Yosida generation theorem. Then a number of well-posednessresults for different notions of a solution (classical/strong/weak) can be derived from the classicalresults for the abstract Cauchy problem [14, 26]. The semigroup theory allows the treatment ofsemilinear problems [11] as well, resulting in the existence and uniqueness result for the semilinear

Kresimir Burazin∗ & Marko Erceg 1


non-stationary Friedrichs system. For some estimates on the solution of such problems we referto [8].

Although the well-posedness result was not known, there was already an aspiration for thenumerical treatment of non-stationary Friedrichs systems [10, 13]. Since a good well-posednesstheorem is desirable for the convergence analysis of a numerical scheme, we hope that our paperwill pave the way for new numerical results in this direction. For example, it would be interestingto see weather results of [22] can be extended to Friedrichs systems, as well.

The paper is organised as follows: in the rest of this introductory section we recall theabstract setting of [19]. The main results are contained in the second section, where we start byintroducing the abstract non-stationary Friedrichs system, then discuss some a priori bounds, andfinally prove that our operator satisfies conditions of the Hille-Yosida theorem. This enables us togive the existence and uniqueness results for different notions of solution. We also prove that theweak solution satisfies the starting equation in a certain vector valued L1 space, and discuss thesemilinear problem. In the third section we apply these result to a number of examples, namelythe Cauchy problem for the symmetric hyperbolic system, the initial-boundary value problemfor the unsteady Maxwell system, the initial-boundary value problem for the unsteady div-gradproblem, and the initial-value problem for the wave equation. At the end, in order to make thepaper easier to read, we have added an Appendix containing basic information regarding theapplication of the semigroup theory to the abstract Cauchy problem.

Abstract Hilbert space formalism

We start by describing the Hilbert space formalism introduced in [19], and recalling some basicresults about the abstract Friedrichs systems. By L we denote a real Hilbert space, identified withits dual L′. Let D ⊆ L be its dense subspace, and L, L : D −→ L (unbounded) linear operatorssatisfying

(T1) (∀ϕ,ψ ∈ D) 〈 Lϕ | ψ 〉L = 〈ϕ | Lψ 〉L ,

(T2) (∃ c > 0)(∀ϕ ∈ D) ‖(L+ L)ϕ‖L 6 c‖ϕ‖L .

These properties together will be refered as (T). By (T1) the operators L and L are closable,

as they have densely defined formal adjoints, with the closures denoted by L and ¯L, and the

corresponding domains by D(L) and D(¯L), respectively.The space (D, 〈 · | · 〉L) is an inner product space, with the graph inner product 〈 · | · 〉L :=

〈 · | · 〉L + 〈 L· | L· 〉L (the corresponding norm ‖ · ‖L is usually called the graph norm), whosecompletion we denote by W0. Analogously we could have defined 〈 · | · 〉L which, by (T2), leadsto the norm that is equivalent to ‖ · ‖L. W0 is continuously imbedded in L (as L is closable), with

the image of W0 equal to D(L) = D(¯L). Moreover, when equipped with the graph norm, thesespaces are isometrically isomorphic.

As L, L : D −→ L are continuous with respect to the graph topology on D, they can beextended by density to a unique operator from L(W0;L) (i.e. a continuous linear operator from

W0 to L). These extensions actually coincide with L and ¯L (taking into account the isomorphismbetween W0 and D(L)). For simplicity, we shall drop the bar from notation and simply writeL, L ∈ L(W0;L), prompted by the fact that (T) still hold for ϕ,ψ ∈W0.

Having in mind the Gelfand triple (the imbeddings are dense and continuous)

W0 → L ≡ L′ →W ′0 ,

the adjoint operator L∗ ∈ L(L;W ′0) defined by

(∀ u ∈ L)(∀ v ∈W0) W ′0〈 L∗u, v 〉W0 = 〈 u | Lv 〉L



satisfies L = L∗|W0

, as (T1) implies

W ′0〈 L∗u, v 〉W0 = 〈 Lu | v 〉L = W ′0

〈 Lu, v 〉W0 , u ∈W0 .

Therefore, the continuous linear operator L : W0 −→ L → W ′0 from (W0, ‖ · ‖L) to W ′0, has aunique continuous extension L∗ to the whole L (the same holds for L∗ and L instead of L andL∗). In order to further simplify the notation we shall write L and L also for their extensions L∗and L∗, respectively.

ByW := u ∈ L : Lu ∈ L = u ∈ L : Lu ∈ L ,

we denote the graph space which, equipped with the graph norm, is an inner product space,containing W0. Furthermore, (W, 〈 · | · 〉T ) is a Hilbert space [19, Lemma 2.1].

In particular, if we take L to be a partial differential operator (see an example below), thanit is clear why the following problem is of interest:

for given f ∈ L find u ∈W such that Lu = f.

To be more precise, the problem is to find sufficient conditions on a subspace V ⊆ W , suchthat the operator L|V : V −→ L is an isomorphism. In the case of a partial differential operator,

the subspace V should contain information about the boundary conditions.In order to find such sufficient conditions, we first introduce a boundary operator D ∈

L(W ;W ′) defined by

W ′〈Du, v 〉W := 〈 Lu | v 〉L − 〈 u | Lv 〉L , u, v ∈W .

It easily follows [19, Lemma 2.3] that D is symmetric:

(∀ u, v ∈W ) W ′〈Du, v 〉W = W ′〈Dv, u 〉W ,

while kerD = W0 [19, Lemma 2.4] justifies the usage of the term boundary operator.

We are now ready to describe V : let V and V be two subspaces of W satisfying

(V 1)(∀ u ∈ V ) W ′〈Du, u 〉W > 0 ,

(∀ v ∈ V ) W ′〈Dv, v 〉W 6 0 ,

(V 2) V = D(V )0 , V = D(V )0 ,

where 0 stands for the annihilator. We shall refer to both (V1) and (V2) as (V). In practice, one

usually starts from a desired V , uses the second equality in (V2) as the definition of V , and thenit remains to check (V1) and the first equality in (V2). The following lemma is immediate.

Lemma 1. If (T) and (V2) hold, then V and V are closed, and kerD = W0 ⊆ V ∩ V .

In order to get the well-posedness result (for the stationary case) one additional assumptionthat ensures the coercivity property of L and L is needed:

(T3) (∃µ0 > 0)(∀ϕ ∈ D) 〈 (L+ L)ϕ | ϕ 〉L > 2µ0‖ϕ‖2L .

It is easy to see that the above property then holds for an arbitrary ϕ ∈ L. The operator Lthat satisfies (T) and (T3) for some L, we call the abstract Friedrichs operator. The proof of thewell-posedness result uses the following lemma.

Lemma 2. ([19, Lemma 3.2]) Under assumptions (T1)–(T3) and (V), the operators L and Lare L–coercive on V and V , respectively; in other words:

(∀ u ∈ V ) 〈 Lu | u 〉L > µ0‖u‖2L ,(∀ v ∈ V ) 〈 Lv | v 〉L > µ0‖v‖2L .



Theorem 1. ([19, Theorem 3.1]) If (T1)–(T3) and (V) hold, then the restrictions of operatorsL|V : V −→ L and L|

V: V −→ L are isomorphisms.

The above theorem gives sufficient conditions on subspaces V and V that ensure well-posedness of the following problems:1) for given f ∈ L find u ∈ V such that Lu = f;

2) for given f ∈ L find v ∈ V such that Lv = f.Its importance also arises from relative simplicity of geometric conditions (V), which, in the casewhen L is a partial differential operator, do not involve the notion of traces for functions in thegraph space of L.

Example. (Classical Friedrichs operator) Let d, r ∈ N, and Ω ⊆ Rd be an open andbounded set with a Lipschitz boundary Γ. Furthermore, assume that the matrix functions Ak ∈W1,∞(Ω; Mr(R)), k ∈ 1..d, satisfy

(F1) Ak is symmetric: Ak = A>k ,

and C ∈ L∞(Ω; Mr(R)). If we denote D := C∞c (Ω; Rr), L = L2(Ω; Rr), and define operatorsL, L : D −→ L by formulæ

Lu :=

d∑k=1

∂k(Aku) + Cu ,

Lu :=−d∑

k=1

∂k(A>k u) + (C> +

d∑k=1

∂kA>k )u ,

where ∂k stands for the classical derivate, then one can easily see that L and L satisfy (T)(for a fixed L, L is defined so that (T1) holds, while (F1) implies (T2)). Therefore, L andL can be uniquely extended to respective operators from L(L;W ′0). Actually, the formulæ forthese extensions remain formally the same as for the starting operators, with the distributionalderivatives taken instead of the classical ones [2].

The graph space W in this example is given by

W =

u ∈ L2(Ω; Rr) :d∑

k=1

∂k(Aku) + Cu ∈ L2(Ω; Rr),

where we have to take distributional derivatives in the above formula. If we denote by ν =(ν1, ν2, . . . , νd) ∈ L∞(Γ; Rd) the unit outward normal on Γ, and define a matrix field on Γ by

Aν :=

d∑k=1

νkAk ,

then for u, v ∈ C∞c (Rd; Rr) the boundary operator D is given by

W ′〈Du, v 〉W =

∫Γ

Aν(x)u|Γ(x) · v|Γ(x)dS(x) .

If we additionally assume

(F2) (∃µ0 > 0) C + C> +d∑

k=1

∂kAk > 2µ0I (a.e. on Ω) ,

then the property (T3) is also satisfied. The inequality above is meant in the sense of the orderon symmetric matrices.



Remark. If we start in the previous example from the beginning with the coefficients defined onthe whole Rd instead of Ω, and consider the operators L, L : D −→ L defined by above formulæ,with D := C∞c (Rd; Rr), L = L2(Rd; Rr), then one can easily see that W = W0 = kerD [1, 7, 24],

which implies that the boundary operator D is trivial. Therefore, we can take V = V = W , andthese spaces satisfy conditions (V), which alows us to consider the initial value problem withoutany other constrains in the non-stationary case.

2. Non-stationary problem

The abstract Cauchy problem

In the rest of the paper we shall consider an initial-boundary value problem for a non-stationary Friedrichs system. To be specific, using the notation from the previous section, weshall consider the abstract Cauchy problem:

(P)

u′(t) + Lu(t) = f

u(0) = u0,

where u : [0, T 〉 −→ L, T > 0, is the unknown function, while the right-hand side f : 〈0, T 〉 −→ L,the initial data u0 ∈ L and the abstract Friedrichs operator L are given. More precisely, we assumethat L does not depend on the time variable t and satisfies (T) , so that it can be extended tothe continuous linear operator L −→W ′0, as in the introductory section. Instead of the positivityassumption (T3), we require that L satisfies a weaker nonnegativity assumption

(T3′) (∀ϕ ∈ D) 〈 (L+ L)ϕ | ϕ 〉L > 0 ,

but such operator we still call the abstract Friedrichs operator. Similarly as it was done in thecase of property (T3), one can see that (T3′) actually holds for arbitrary ϕ ∈ L.

Remark. In reference to the classical Friedrichs operator of the example above, the condition(F2) can be weakened to

(F2′) C + C> +d∑

k=1

∂kAk > 0 (a.e. on Ω) ,

as this is enough to ensure the validity of (T3′).Actually, if L satisfies only (F1), the positivity condition (F2) can be achieved by substituting

v := e−λtu, for some suitable λ > 0: then the corresponding non-stationary system becomes

∂tv + (L+ λI)v = e−λtf ,

where I is the identity matrix, and since all matrices appearing in the above expression arebounded, we can choose λ large enough so that (F2) is satisfied. Therefore, the operator L+ λIand its formal adjoint satisfy (T1)–(T3). Note also that the initial condition u(0, ·) = u0 isequivalent to v(0, ·) = u0 while the graph space of operator L+λI is the same as the graph spaceof L, with equivalent norms.

In order to fully describe problem (P) (see Appendix), it remains to be clarified what is thedomain D(L) of L. As the equation (P)1 should hold in L, we can take any subspace of thegraph space W . If L is a partial differential operator, then the choice of domain of L correspondsto given boundary conditions, and it is then natural to take for D(L) to be some subspace Vsatisfying (V). Then one can easily prove a similar result as in Lemma 2 if the weaker assumption(T3′) is taken instead of (T3).



Lemma 3. Under assumptions (T), (T3′) and (V), the operators L and L are L-accretive on

V and V , respectively, i.e. they satisfy

(∀ u ∈ V ) 〈 Lu | u 〉L > 0 ,

(∀ v ∈ V ) 〈 Lv | v 〉L > 0 .

If u is the classical solution of (P) on [0, T 〉 (see the Appendix for definitions of differentnotions of a solution), one can easily derive an a priori estimate

(∀ t ∈ [0, T ]) ‖u(t)‖2L 6 et(‖u0‖2L +

∫ t

0‖f(s)‖2L

),

from which the uniqueness of classical solution is immediate. Indeed, from (P)1 we have

〈 u′(t) | u(t) 〉L + 〈 Lu(t) | u(t) 〉L = 〈 f(t) | u(t) 〉L ,

and by (T3′) it follows

d

dt‖u(t)‖2L 6 2〈 f(t) | u(t) 〉L 6 ‖u(t)‖2L + ‖f(t)‖2L ,

from where, after applying the differential form of the Gronwall inequality, we get the a prioriestimate. Here we have assumed that the right-hand side of (P)1 is square-integrable, i.e. f ∈L2(〈0, T 〉;L). Clearly, this is not sufficient for the existence of the classical solution and precisesufficient assumptions will be presented in the sequel, as well as a sharper a priori estimate.

Semigroup approach

In order to fit our problem (P) in the setting of the semigroup theory as presented in theAppendix, let us take a subspace V of W satisfying (V), and define an operator A : V −→ L byA := −L|V . We can now rewrite our problem (P) in terms of operator A:

(P′)

u′(t)−Au(t) = f

u(0) = u0,

and since the domain V of A is equipped with the topology inherited from L (and not the graphnorm), A is unbounded in general.

Our goal is to show that A is an infinitesimal generator of a strongly continuous semigroup(C0-semigroup) on L. If we succeed, then our problem (P′) (and thus also (P)) can be fitted inthe setting of abstract Cauchy problems of the semigroup theory. The following theorem is themain result of this paper.

Theorem 2. A is the infinitesimal generator of a contraction C0-semigroup (T (t))t>0 on L.

Dem. According to the Hille-Yosida generation theorem (Theorem 7) it is necessary and sufficientto show thata) V is dense in L,b) A is closed,c) ρ(A) ⊇ 〈0,∞〉 and ‖Rλ(A)‖ 6 1

λ , for every λ > 0.Density of V is a direct consequence of Lemma 1. In order to prove (b), let us take a sequence

(un) in V such that un −→ u and Aun −→ f, both in L. We need to show that u ∈ V and f = Au.It is easy to show that (un) is a Cauchy sequence in the complete graph space W . Indeed,

‖un − um‖2W = ‖un − um‖2L + ‖Lun − Lum‖2L= ‖un − um‖2L + ‖Aun −Aum‖2L ,



and the claim follows by the assumption that (un) and (Aun) are convergent (therefore Cauchy)sequences in L. Thus (un) is convergent in W and, because V is closed in W , we additionally havethat the limit v is in V . The convergence in the graph norm gives us un −→ v and Aun −→ Avin L, which implies u = v ∈ V and f = Av = Au.

In order to prove the last statement, let us define Lλ := λI + L and a corresponding Lλ :=λI+ L, for an arbitrary λ > 0, and let us show that Lλ is an isomorphism from V to L. The ideaof proof is to show that Lλ and Lλ satisfy (T1)–(T3) and then apply Theorem 1. The condition(T1) is trivially satisfied, (T2) is satisfied if we take c + 2λ as the constant, and since L and Lsatisfy (T3′) we have

〈 (Lλ + Lλ)u | u 〉L = 〈 (L+ L)u | u 〉L + 2λ〈 u | u 〉L > 2λ‖u‖2L ,

for every u ∈ L, which is exactly (T3). Before applying Theorem 1, it is important to notice thatthe corresponding graph space Wλ and the boundary operator Dλ of operator Lλ do not dependon λ. Indeed, one can verify that for every λ > 0 we have Wλ = W and Dλ = D, which impliesthat subspace V satisfy conditions (V) in the graph space of operator Lλ. Now we have thatLλ|V : V −→ L is an isomorphism, and therefore ρ(A) ⊇ 〈0,∞〉. Finally, Lemma 2 gives us

‖Lλ|V ‖ > λ =⇒ ‖Rλ(A)‖ 6 1

λ,

because Rλ(A) = (Lλ|V )−1.

Q.E.D.

Remark. Instead of using the Hille-Yosida theorem in the proof of the previous theorem, wecould alternatively (like in [14, p. 412] and [26, p. 14]) prove that A is maximal dissipative, i.e.A is dissipative (which is a direct consequence of (T3′)) and im (I −A) = L, and then apply theLumer-Phillips theorem.

After we have established that A generates a contraction C0-semigroup we can use someclassical results (see Appendix for some of them, and [11, 26] for a more extensive presentation)to derive various conclusions about solvability of (P). In the following corollary we shall summariseresults that follow from the results stated in the Appendix.

Corollary 1. Let L be an operator that satisfies (T) and (T3′), V a subspace of its graph spacesatisfying (V), and f ∈ L1(〈0, T 〉;L).a) Then for every u0 ∈ L the problem (P) has the unique weak solution u ∈ C([0, T ];L) given

by

u(t) = T (t)u0 +

∫ t

0T (t− s)f(s)ds , t ∈ [0, T ] ,

where (T (t))t>0 is as in Theorem 2.

b) If additionally u0 ∈ V and f ∈W1,1(〈0, T 〉;L)∪(

C([0, T ];L)∩L1(〈0, T 〉;V ))

, with V equipped

with the graph norm, then the above weak solution is the classical solution of (P) on [0, T 〉.

Remark. From the formula for the solution one can easily get the estimate

(∀ t ∈ [0, T ]) ‖u(t)‖L 6 ‖u0‖L +

∫ t

0‖f(s)‖Lds .

By the definition of weak solution it is clear that it satisfies the initial condition, while theequation is satisfied in the meaning of Theorem 10. However, in our setting we can show thatthe weak solution satisfies the equation in a certain space (a similar observation has been donein [14, p. 406]).



Theorem 3. Let u0 ∈ L, f ∈ L1(〈0, T 〉;L) and let u be the weak solution of (P). Then u′,Lu, f ∈L1(〈0, T 〉;W ′0) and

u′ + Lu = f ,

in L1(〈0, T 〉;W ′0).

Dem. According to Theorem 10, there exists a sequence of the classical solutions un to (P′)with right hand sides fn ∈ C1([0, T ];L) and initial conditions u0n ∈ V , such that fn −→ f inL1(〈0, T 〉;L), u0n −→ u0 in L, and un −→ u in C([0, T ];L).

Since L ∈ L(L;W ′0), we have Lun −→ Lu in C([0, T ];W ′0), which implies the convergencein the space L1(〈0, T 〉;W ′0). As L is continuously embedded in W ′0, we also have fn −→ f inL1(〈0, T 〉;W ′0). Thus

u′n = −Lun + fn −→ −Lu + f ,

in L1(〈0, T 〉;W ′0). On the other hand un −→ u in the space of vector valued distributionsD′(〈0, T 〉;W ′0), which implies u′n −→ u′ in D′(〈0, T 〉;W ′0) (see [14, p. 218]). The uniquenessof the limit in the space of vector valued distributions gives us

u′ + Lu = f

in D′(〈0, T 〉;W ′0), and since Lu, f ∈ L1(〈0, T 〉;W ′0) and L1(〈0, T 〉;W ′0) is continuously embeddedin D′(〈0, T 〉;W ′0), we also have u′ ∈ L1(〈0, T 〉;W ′0).

Q.E.D.

Remark. As it is well known, semilinear problems can also be treated via the semigroup theory[11, 26]. Therefore, we can get the existence and uniqueness result for the abstract Cauchyproblem

u′(t) + Lu(t) = f(t, u(t))

u(0) = u0,

where L is an abstract Friedrichs operator, and the right-hand side f : 〈0, T 〉 × L −→ L dependsalso on the unknown function. Usual assumptions on f that ensure existence and uniqueness ofthe weak solution are continuity and some kind of Lipschitz continuity in the last variable (see[11, Ch. 4] and [26, Ch. 6]). For some estimates on the solution see [8].

3. Examples

We shall now prove that some well-known examples fit in our setting of non-stationaryFriedrichs systems, and apply the results of the previous section.

Initial problem for symmetric hyperbolic system

We consider the initial problem for first-order system

(HS)

∂tu +

d∑k=1

∂k(Aku) + Cu = f in 〈0, T 〉 ×Rd

u(0, ·) = u0

,

where T > 0 is some given time, and the coefficients Ak ∈ W1,∞(Rd; Md(R)), k ∈ 1..n, C ∈L∞(Rd; Md(R)) do not depend on time. For the right-hand side and the initial datum we supposef ∈ L1(〈0, T 〉; L2(Rd; Rd)) and u0 ∈ L2(Rd; Rd), while u : [0, T 〉 × Rd −→ Rd is the unknownfunction. We also suppose that each Ak is symmetric a.e. on Rd

One can easily see that (HS) fits in our framework of non-stationary Friedrichs systems withthe operator L taken to be

Lu :=d∑

k=1

∂k(Aku) + Cu .



Indeed, the symmetry condition (F1) is trivially satisfied, while the positivity condition (F2) canbe obtained by using the substitution v := e−λtu, for some λ > 0 large enough, as remarkedbefore. Then our Friedrichs system reads

∂tv + (L+ λI)v = e−λtf ,

the initial condition remains v(0, ·) = u0, and the graph space of the operator L+ λI is the sameas the graph space of L, with equivalent norms: W = w ∈ L2(Rd; Rd) : Lw ∈ L2(Rd; Rd).Now, as a consequence of the remark closing the introductory section and Corollary 1, we havethe result of existence and uniqueness of the solution to (HS).

Theorem 4. Let f ∈ W1,1(〈0, T 〉; L2(Rd; Rd)) ∪(

C([0, T ]; L2(Rd; Rd)) ∩ L1(〈0, T 〉;W ))

and

u0 ∈W . Then the abstract initial-value problem u′ +d∑

k=1

∂k(Aku) + Cu = f

u(0) = u0

has the unique classical solution given by

u(t) = eλtT(t)u0 +

∫ t

0eλ(t−s)T(t− s)f(s)ds , t ∈ [0, T ] ,

where (T(t))t>0 is the contraction C0-semigroup generated by −L− λI.

Remark. The above formula could have been written in terms of the C0-semigroup S(t) :=eλtT(t), with the infinitesimal generator −L [26, p. 12].

Remark. A similar existence and uniqueness result for the Cauchy problem for the symmetrichyperbolic systems, under some additional assumptions on coefficients can be found in [20].

Time-dependent Maxwell system

Let Ω ⊆ R3 be open and bounded with a Lipschitz boundary Γ, T > 0, Σij ∈ L∞(Ω; M3(R)),i, j ∈ 1, 2, f1, f2 ∈ L1(〈0, T 〉; L2(Ω; R3)), µ, ε ∈ W1,∞(Ω; M3(R)) and constants µ0, ε0 ∈ R+

such that for every x ∈ Ω, µ(x) and ε(x) are symmetric, and µ > µ0I, ε > ε0I. We consider thegeneralised time-dependent Maxwell system:

(MS)

µ∂tH + rot E + Σ11H + Σ12E = f1

ε∂tE− rot H + Σ21H + Σ22E = f2in 〈0, T 〉 × Ω ,

where E,H : [0, T 〉 × Ω −→ R3 are the unknown functions.

Remark. If we take f1 ≡ 0, f2 = J, Σ11 = Σ12 = Σ21 ≡ 0 we will get the (standard) Maxwellsystem in a linear nonhomogeneous anisotropic medium. In that case H, E, ε, Σ22, µ, J representthe magnetic and electric fields, the electric permeability, the conductivity of the medium, themagnetic susceptibility of the material in Ω and the applied current density, respectively.

The principal square roots µ(x)12 and ε(x)

12 of µ(x) and ε(x) are well-defined for every x ∈ Ω

because µ(x) and ε(x) are positive-definite matrices, which justifies definitions µ12 (x) := µ(x)

12 ,

ε12 (x) := ε(x)

12 . Moreover, we have µ

12 , ε

12 ∈ W1,∞(Ω; M3(R)), both µ

12 (x) and ε

12 (x) are

symmetric, for every x ∈ Ω, with µ12 > µ0

12 I, ε

12 > ε0

12 I. The last property implies that

µ12 (x) and ε

12 (x) are regular, for every x, and moreover µ−

12 , ε−

12 ∈ W1,∞(Ω; M3(R)), where

µ−12 (x) := µ

12 (x)−1 and ε−

12 (x) := ε

12 (x)−1.



After introducing the substitutions (using the previous notation)

u :=

[u1

u2

]=

[µ

12 H

ε12 E

],

and multiplying the first three equations by µ−12 , and the last three by ε−

12 , we can write our

system as

∂tu + Lu = F ,

where

Lu :=

[µ−

12 rot (ε−

12 u2)

−ε−12 rot (µ−

12 u1)

]+

[µ−

12 Σ11µ

− 12 µ−

12 Σ12ε

− 12

ε−12 Σ21µ

− 12 ε−

12 Σ22ε

− 12

] [u1

u2

], F =

[µ−

12 f1

ε−12 f2

].

To conclude that L is a classical Friedrichs operator we still need to do some computations:

µ−12 rot (ε−

12 u2) = µ−

12

3∑k=1

∂k(Bkε− 1

2 u2)

=

3∑k=1

∂k(µ− 1

2Bkε− 1

2 u2)−( 3∑k=1

(∂kµ− 1

2 )Bkε− 1

2

)u2 ,

where

B1 =

0 0 00 0 −10 1 0

, B2 =

0 0 10 0 0−1 0 0

, B3 =

0 −1 01 0 00 0 0

are matrices corresponding to the differential operator rot . We can analougously rewrite the last

three components −ε−12 rot (µ−

12 u1) and get:

[µ−

12 rot (ε−

12 u2)

−ε−12 rot (µ−

12 u1)

]=

3∑k=1

∂k

([0 µ−

12Bkε

− 12

ε−12B>k µ

− 12 0

]u

)

+

(3∑

k=1

[0 (∂kµ

− 12 )B>k ε

− 12

(∂kε− 1

2 )Bkµ− 1

2 0

])u .

If we define

Ak :=

[0 µ−

12Bkε

− 12

ε−12B>k µ

− 12 0

], k ∈ 1..3 ,

C :=3∑

k=1

[0 (∂kµ

− 12 )B>k ε

− 12

(∂kε− 1

2 )Bkµ− 1

2 0

]+

[µ−

12 Σ11µ

− 12 µ−

12 Σ12ε

− 12

ε−12 Σ21µ

− 12 ε−

12 Σ22ε

− 12

],

then one can easily see that L takes the form

Lu =

3∑k=1

∂k(Aku) + Cu

of the classical Friedrichs partial differential operator. Indeed, the symmetry condition (F1)is trivially satisfied, while the positivity condition (F2) can be obtained using the substitutionv := e−λtu, as in the previous example. Therefore, without loss of generality, let us assume thatour operator L satisfies the positivity condition (F2).



Remark. As the system above is a symmetric hyperbolic system, the Cauchy problem on wholeR3 is contained in the setting of the previous example, so the existence and uniqueness result isimmediate. Using that, we can get the existence and uniqueness result for the Cauchy problemof the starting Maxwell system (MS).

Obviously, the spaces involved are

L = L2(Ω; R3)× L2(Ω; R3) ,

W =

u =

[u1

u2

]∈ L :

[rot (µ−

12 u1)

rot (ε−12 u2)

]∈ L

=

u =

[u1

u2

]∈ L :

[µ−

12 u1

ε−12 u2

]∈ L2

rot(Ω; R3)× L2rot(Ω; R3)

,

where L2rot(Ω; R3) stands for the graph space of differential operator rot . Because µ−

12 and ε−

12

are uniformely bounded from below and above the graph norm ‖ · ‖L is equivalent with the norm

‖u‖ :=

∥∥∥∥∥[µ−

12 u1

ε−12 u2

]∥∥∥∥∥L2rot(Ω;R3)×L2

rot(Ω;R3)

on W .If we denote by ν = (ν1, ν2, ν3) ∈ L∞(Γ; R3) the unit outward normal on Γ, matrix field Aν

can be written as a block matrix

Aν =

[0 µ−

12 Arot

ν ε−12

−ε−12 Arot

ν µ−12 0

],

where

Arotν =

0 −ν3 ν2

ν3 0 −ν1

−ν2 ν1 0

is a matrix corresponding to the operator rot . We can describe the boundary operator D in terms

of the trace operator TH1 : H1(Ω; R3) −→ H12 (Γ; R3) and the tangential trace operator (see [1],

[7]) Trot : L2rot(Ω; R3) −→ H−

12 (Γ; R3). Indeed, for u, v ∈ C∞c (R3; R6), we have

W ′〈Du, v 〉W =

∫Γ

Aν(x)u|Γ(x) · v|Γ(x)dS(x)

=H−

12〈µ−

12Trot(ε

− 12 u2), TH1v1 〉

H12−

H−12〈 ε−

12Trot(µ

− 12 u1), TH1v2 〉

H12

=H−

12〈µ−

12Trot(ε

− 12 v2), TH1u1 〉

H12−

H−12〈 ε−

12Trot(µ

− 12 v1), TH1u2 〉

H12,

where the second equation can be extended by density to (u, v) ∈W ×H1(Ω; R6), and the thirdone to (u, v) ∈ H1(Ω; R6)×W . Also note that in the above expression we have multiplication of

a functional from H−12 (Γ; R3) by matrix Lipschitz functions µ−

12 and ε−

12 , which is well-defined

as follows

H−12〈µ−

12 g, z 〉

H12

:=H−

12〈 g,µ

− 12

|Γz 〉

H12, z ∈ H

12 (Γ; R3) , g ∈ H−

12 (Γ; R3) ,

and analogously for ε−12 .

Let us take the subspace V that corresponds to the boundary condition ν × E|Γ = 0 (in the

case when the corresponding function is not smooth, we interpret this notion in terms of the traceoperators). Now we take

V = V = u ∈W : Trot(ε− 1

2 u2) = 0= u ∈W : TrotE = 0



and verify the conditions (V1) and (V2). Note that u ∈ V if and only if [H E]> ∈ L2rot(Ω; R3)×

L2rot,0(Ω; R3), where L2

rot,0(Ω; R3) := Cl L2rot(Ω;R3)C

∞c (Ω; R3).

To prove (V1) we need to show

(∀ u ∈ V ) W ′〈Du, u 〉W = 0 .

Since u = [u1 u2]> ∈ V implies µ−12 u1 ∈ L2

rot(Ω; R3) and ε−12 u2 ∈ L2

rot,0(Ω; R3), there is a

sequence (un) = ([un1 un2 ]>) in H1(Ω; R3) × C∞c (Ω; R3) such that un −→ u in W . As TH1un2 = 0

and Trot(ε− 1

2 un2 ) = ν ×TH1(ε−12 un2 ) = 0, using computed identity for D and its continuity, we get

W ′〈Du, u 〉W = limn

W ′〈Dun, un 〉W = limn

0 = 0 ,

which proves (V1).In order to prove (V2), let us first take arbitrary u, v ∈ V and the corresponding approxi-

mating sequences (un), (vn) from H1(Ω; R3)×C∞c (Ω; R3), as in the previous case. Similarly likebefore, we have

W ′〈Dv, u 〉W = limn

W ′〈Dvn, un 〉W = limn

0 = 0 ,

as Trot(ε− 1

2 un2 ) = TH1vn2 = 0, which gives us V ⊆ D(V )0. For the opposite inclusion we need toshow that for an arbitrary u = [u1 u2]> ∈W , condition

(∀ v ∈ V ) W ′〈Dv, u 〉W = 0 ,

implies Trot(ε− 1

2 u2) = 0. If we put in the above equality v = [v1 0]>, v1 ∈ H1(Ω; R3), we get

W ′〈Dv, u 〉W =H−

12〈µ−

12Trot(ε

− 12 u2), TH1v1 〉

H12

= 0 .

Since v1 ∈ H1(Ω; R3) is arbitrary, TH1 is surjective and the mapping z 7→ µ−12 z is from H

12 (Γ; R3)

onto H12 (Γ; R3), we get, Trot(ε

− 12 u2) = 0, and (V2) is obtained.

Let us summarise, using Corollary 1, all that has been shown in the following theorem.

Theorem 5. Let E0 ∈ L2rot,0(Ω; R3),H0 ∈ L2

rot(Ω; R3) and let f1, f2 ∈ C([0, T ]; L2(Ω; R3)) satisfyat least one of the following two conditions

i) f1, f2 ∈W1,1(〈0, T 〉; L2(Ω; R3));ii) µ−1f1 ∈ L1(〈0, T 〉; L2

rot(Ω; R3)), ε−1f2 ∈ L1(〈0, T 〉; L2rot,0(Ω; R3)).

Then the abstract initial-boundary value problem

µH′ + rot E + Σ11H + Σ12E = f1

εE′ − rot H + Σ21H + Σ22E = f2

E(0) = E0

H(0) = H0

ν × E|Γ = 0

has the unique classical solution given by[HE

](t) =

[µ−

12 0

0 ε−12

]T(t)

[µ

12 H0

ε12 E0

]+

[µ−

12 0

0 ε−12

] ∫ t

0T(t− s)

[µ−

12 f1(s)

ε−12 f2(s)

]ds , t ∈ [0, T ] ,

where (T(t))t>0 is the contraction C0-semigroup generated by −L.

Remark. The above theorem is valid if the corresponding Friedrichs operator satisfies (F2′).As written before, if this is not the case, then the positivity condition can be obtained by a simplechange of variable. The statement of the above theorem should also be changed accordingly.



We can get a similar result if we take the boundary condition ν × H|Γ = 0 instead of

ν × E|Γ = 0. More precisely, it can be shown that spaces

V = V = u ∈W : Trot(µ− 1

2 u1) = 0= u ∈W : TrotH = 0

satisfy (V), which gives us well-posedness of the corresponding initial-boundary value problem.

Remark. For more information regarding the Maxwell system, we refer to [12], [14].

Unsteady div-grad problem

Let Ω ⊆ Rd be open and bounded with a Lipschitz boundary Γ, f1 ∈ L1(〈0, T 〉; L2(Ω; Rd)),f2 ∈ L1(〈0, T 〉; L2(Ω; R)), and c > 0. We consider a system of equations

(DG)

∂tq +∇p = f11

c2∂tp+ div q = f2

in 〈0, T 〉 × Ω ,

with the unknowns p : [0, T 〉 × Ω −→ R and q : [0, T 〉 × Ω −→ Rd.

Remark. These are linearised equations for the propagation of sound in the inviscid, elastic andcompressible fuid, describing small disturbances [25]. Here, the unknowns q and p correspond tothe velocity of the fluid and the pressure, while the constant c stands for the speed of sound inthe fluid.

After introducing the substitution

u :=

[u1

u2

]=

[1cpq

],

we get an evolution Friedrichs system

∂tu + Lu = F ,

where F =

[cf2

f1

]and Lu :=

[c div u2

c∇u1

]is the classical Friedrichs operator satisfying (F1) and

(F2′), with C = 0 and Ak = c e1 ⊗ ek+1 + c ek+1 ⊗ e1 ∈ M1+d(R).Obviously, we have

W = H1(Ω)× L2div(Ω; Rd) ,

W0 = H10(Ω)× L2

div,0(Ω; Rd) = ClWC∞c (Ω; R1+d) ,

where L2div(Ω; Rd) is the graph space of differential operator div , and L2

div,0(Ω; Rd) is the closure

of C∞c (Ω; Rd) in the graph norm of L2div(Ω; Rd).

Remark. As the system above is a symmetric hyperbolic system, the Cauchy problem onwhole Rd is contained in the setting of the first example, so the existence and uniqueness resultis immediate, which also implies the existence and uniqueness result for the Cauchy problem ofthe unsteady div-grad problem. In this case we have V = V = W = H1(Rd)× L2

div(Rd; Rd).

The matrix field Aν can be written as a block matrix

Aν =

[0 cν>

cν 0

],



where ν is the unit outward normal on Γ, as before, and the boundary operator D can be described

in terms of the trace operator TH1 : H1(Ω) −→ H12 (Γ) and the normal trace operator (see [1], [7])

Tdiv : L2div(Ω; Rd) −→ H−

12 (Γ):

W ′〈Du, v 〉W =

∫Γ

Aν(x)u|Γ(x) · v|Γ(x)dS(x)

= cH−

12〈 Tdivu2, TH1v1 〉

H12

+ cH−

12〈 Tdivv2, TH1u1 〉

H12,

for u, v ∈ C∞c (Ω; R1+d). The second equality is actually valid for all (u, v) ∈ W ×W , by thedensity argument.

We shall study two possible boundary conditions, the first one is given by

V = V = u ∈W : TH1u1 = 0= u ∈W : TH1p = 0= H1

0(Ω)× L2div(Ω; Rd) ,

and it clearly corresponds to the boundary condition p|Γ = 0. For the subspace V we can check

the condition (V1) analogously as in the previous example, as well as V ⊆ D(V )0. For the proofof the opposite inclusion in (V2) we need to show that for an arbitrary u ∈W , the condition

(∀ v ∈ V ) W ′〈Dv, u 〉W = 0

implies TH1u1 = 0. If we put in the above equality v = [0 v2]>, v2 ∈ L2div(Ω; Rd), we get

W ′〈Dv, u 〉W = cH−

12〈 Tdivv2, TH1u1 〉

H12

= 0 ,

which implies TH1u1 = 0, by the arbitrariness of v2 ∈ L2div(Ω; Rd) and the surjectivity of Tdiv.

Using Corollary 1, we can summarise the above computations in the following theorem.

Theorem 6. Let q0 ∈ L2div(Ω; Rd), p0 ∈ H1

0(Ω) and let functions f1 ∈ C([0, T ]; L2(Ω; Rd)) andf2 ∈ C([0, T ]; L2(Ω)) satisfy at least one of the following two conditions

i) f1 ∈W1,1(〈0, T 〉; L2(Ω; Rd)), f2 ∈W1,1(〈0, T 〉; L2(Ω));ii) f1 ∈ L1(〈0, T 〉; L2

div(Ω; Rd)), f2 ∈ L1(〈0, T 〉; H10(Ω)).

Then the abstract initial-boundary-value problem

q′ +∇p = f11

c2p′ + div q = f2

q(0) = q0

p(0) = p0

p|Γ = 0

has the unique classical solution given by[pq

]=

[c 0>

0 I

]T(t)

[1cp0

q0

]+

[c 0>

0 I

] ∫ t

0T(t− s)

[cf2

f1

]ds , t ∈ [0, T ] ,

where (T(t))t>0 is the contraction C0-semigroup generated by −L.

We can get a similar result for the another possible boundary condition ν · q = 0|Γ , i.e. for

the subspacesV = V = u ∈W : Tdivu2 = 0

= u ∈W : Tdivq = 0= H1(Ω)× L2

div,0(Ω; Rd) .



Initial-value problem for the wave equation

Let us now consider the initial-value problem

(WE)

∂ttu− c24u = f in 〈0, T 〉 ×Rd

u(0, ·) = u0

∂tu(0, ·) = u10

,

where T > 0 is a given constant, c > 0 represents the propagation speed of the wave, f ∈L1(〈0, T 〉; L2(Rd)) is the external force, u0 ∈ H1(Rd) and u1

0 ∈ L2(Rd) are the inital position andvelocity respectively, while u : [0, T 〉 ×Rd −→ R is the unknown function which represents thedisplacement.

After a change of variable

u :=

[u1

u2

]=

[∂tu−c∇u

],

we get the same system ∂tu + Lu = F as in the previous example with f1 ≡ 0 and f2 = 1cf . Note

that the last d equations of this system are actually the Schwarz symmetry relations, while thefirst one results from the wave equation we started with. The initial condition transforms to

u(0) =

[u1(0)u2(0)

]=

[u1

0

−c∇u0

]=: u0 .

Our problem fits in the framework of the Cauchy problem for the unsteady div-grad problem(which is remarked in the previous example), so for V = V = W = H1(Rd) × L2

div(Rd; Rd) wehave the existence result.

However, due to our change of variable, we only have information about the derivatives ofthe original unknown. Therefore, in order to find u one must solve the problem

(odeP)

u′(t) = u1(t)

u(0) = u0.

Let us illustrate this in the case of the classical solution; if we additionally assume f ∈W1,1(〈0, T 〉; L2(Rd))∪

(C([0, T ]; L2(Rd))∩L1(〈0, T 〉; H1(Rd))

), u1

0 ∈ H1(Rd) and 4u0 ∈ L2(Rd),

then (by Corollary 1) the abstract Cauchy problem

(P)

u′(t) + Lu(t) = F

u(0) = u0

has the classical solution u ∈ C1(〈0, T 〉; L2(Rd; Rd+1)) ∩ C([0, T ]; L2(Rd; Rd+1)) ∩ C(〈0, T 〉;W ).Therefore

u1 ∈ C1(〈0, T 〉; L2(Rd)) ∩ C([0, T ]; L2(Rd)) ∩ C(〈0, T 〉; H1(Rd)) ,

u2 ∈ C1(〈0, T 〉; L2(Rd; Rd)) ∩ C([0, T ]; L2(Rd; Rd)) ∩ C(〈0, T 〉; L2div(Rd; Rd)) ,

which ensures that the problem (odeP) has the unique solution u belonging to

C2(〈0, T 〉; L2(Rd)) ∩ C1(〈0, T 〉; H1(Rd)) ∩ C([0, T ]; H1(Rd)) .

Note that this u already satisfies the first initial condition in (WE). Let us show that it alsosatisfies the wave equation in C(〈0, T 〉; L2(Rd)).

Since ∇ : H1(Rd) −→ L2(Rd) is a continuous linear operator, it follows that the opera-tor (which we denote the same) defined by (∇u)(t) := ∇(u(t)) is a linear continuous operatorC1(〈0, T 〉; H1(Rd)) −→ C1(〈0, T 〉; L2(Rd)) and C(〈0, T 〉; H1(Rd)) −→ C(〈0, T 〉; L2(Rd)), whichcommutes with the time derivative: ∇(u′) = (∇u)′. Using this, the last d equations in (P)and (odeP)1, we get u′2 = (−c∇u)′ in the space C(〈0, T 〉; L2(Rd; Rd)), which together with theinitial conditions for u2 and u imply u2 = −c∇u. Note that, as an additional consequence wehave ∇u ∈ C(〈0, T 〉; L2

div(Rd)). Substituting u1 and u2 in the first equation in (P) we get (inC(〈0, T 〉; L2(Rd)))

u′′ = u′1 = −c div u2 + f = −c div (−c∇u) + f = c24u+ f .

It remains to prove that u satisfies the second initial condition in (WE), which easily followsfrom u′ = u1 in C(〈0, T 〉; L2(Rd)), u1 ∈ C([0, T ]; L2(Rd)) and u1(0) = u1

0.



Remark. A similar existence and uniqueness result for the initial-value problem for the waveequation within the framework of a semigroup theory can be found in [26].

Remark. Using results of the unsteady div-grad problem, we can get the existence anduniqueness results for the initial-boundary value problem for the wave equation with two possibleboundary conditions:

∂tu|Γ = 0 ,

which corresponds to the Dirichlet boundary conditions, or

ν · ∇u|Γ = 0 .

which is the homogenous Neumann boundary condition.

Appendix

Semigroups

In this appendix we summarise some basic facts about semigroup theory, which we haveused in the paper (details can be found in [11], [26] and [14, Ch. XVII]). Throughout the section(X, ‖ · ‖X) will denote a real Banach space and ‖ · ‖ the norm on the space of bounded linearoperators on X, L(X).

A family (T (t))t>0 of bounded linear operators on X is called strongly continuous semigroup(C0-semigroup) if it satisfies

i) T (0) = I (identity),ii) (∀ t, s > 0) T (t+ s) = T (t)T (s),

iii) (∀x ∈ X) t 7→ T (t)x is continuous.As a consequence of the uniform boundedness theorem, one can show [26, p. 4] that for every

C0-semigroup (T (t))t>0 there exist constants ω > 0 and M > 1 such that

(∀ t > 0) ‖T (t)‖ 6Meωt .

If in the above expresion we can take ω = 0 and M = 1, then (T (t))t>0 is called a contractionC0-semigroup.

The infinitesimal generator of a C0-semigroup (T (t))t>0 is a map A : D(A) −→ X defined by

Ax := limt0

1

t

(T (t)x− x

),

on in its domain D(A) consisting of those x ∈ X for which the above limit exists. It can beshown that A is a densely defined closed linear operator.

In applications, it is often of the interest to clarify whether a given linear operator A : D(A) ⊆X −→ X is an infinitesimal generator of some C0-semigroup, and if it is, whether we can findan explicit formula for this semigroup. For the latter question, in general situations there are nocomputable formulæ, but for the first one the answer is given by the famous Hille-Yosida theorem.

Theorem 7. (Hille-Yosida) [26, p. 8] A linear operator A : D(A) ⊆ X −→ X is theinfinitesimal generator of some contraction C0-semigroup (T (t))t>0 if and only if

i) D(A) is dense in X,ii) A is closed,

iii) ρ(A) ⊇ 〈0,∞〉 and for every λ > 0 we have ‖Rλ(A)‖ 6 1λ ,

whereρ(A) := λ ∈ R : λI −A : D(A) −→ X is isomorphism

is the resolvent set of A, while

Rλ(A) := (λI −A)−1 , λ ∈ ρ(A) ,

is the resolvent of A.



The abstract Cauchy problem

One of the most prominent applications of the semigroup theory is in solving the abstractCauchy problem

(ACP)

u′(t) +Au(t) = f

u(0) = u0,

where a linear operator A : D(A) ⊆ X −→ X, u0 ∈ X and f : 〈0, T 〉 −→ X, T > 0 are given.We say that u is a classical solution of (ACP) on [0, T 〉 if u ∈ C1(〈0, T 〉;X) ∩ C([0, T ];X),

u(t) ∈ D(A), for 0 < t < T , and u satisfies (ACP). Note that if u is a classical solution of (ACP)and f ∈ C(〈0, T 〉;X), then Au ∈ C(〈0, T 〉;X) (which implies u ∈ C(〈0, T 〉;D(A)) with D(A)equipped with the graph norm), so (ACP)1 is satisfied in the classical sense, i.e. in C(〈0, T 〉;X).However, f ∈ C([0, T ];X) is not sufficient for the existence of a classical solution to (ACP) [26,p. 106], and one needs some stronger assumptions on f in order to get the existence.

We shall first consider the homogeneous case. The result below illustrates the importance ofthe Hille-Yosida theorem in obtaining the existence and uniqueness result for (ACP).

Theorem 8. [26, p. 102] Let A be a densely defined operator with a nonempty resolvent setρ(A) and let f ≡ 0. The problem (ACP) has, for every u0 ∈ D(A), the unique classical solutionu on [0,∞〉 if and only if A is the infinitesimal generator of a C0-semigroup (T (t))t>0. Moreover,u(t) = T (t)u0 for every t > 0.

If the semigroup from the previous theorem is additionally a contraction C0-semigroup, thenfrom the formula for the solution we get the estimate

(∀ t > 0) ‖u(t)‖X 6 ‖u0‖X ,

which means that our problem is dissipative.Let us now consider the nonhomogeneous case, additionally assuming in the sequel that A is

the infinitesimal generator of a C0-semigroup (T (t))t>0. If f ∈ C([0, T ];X) and u0 ∈ D(A), thenone can easily see that any classical solution u of (ACP) satisfies

(CS) u(t) = T (t)u0 +

∫ t

0T (t− s)f(s)ds , t ∈ [0, T ] ,

which proves the uniqueness of classical solution. If (T (t))t>0 is a contraction C0-semigroup, thenwe also have the estimate

(∀ t ∈ [0, T ]) ‖u(t)‖X 6 ‖u0‖X +

∫ t

0‖f(s)‖Xds .

However, in order to check that the right-hand side of (CS) really defines a solution, one needsto verify that the last term above is (continuously) differentiable [26, p. 107] which requires someadditional assumptions on f.

Theorem 9. [11, p. 51] Let f ∈ C([0, T ];X), u0 ∈ D(A), and assume that at least one of thefollowing conditions is satisfied:

i) f ∈W1,1(〈0, T 〉;X);ii) f ∈ L1(〈0, T 〉;D(A)), with D(A) equipped with the graph norm.

Then (ACP) has the unique classical solution u given by (CS).

Note that (CS) has a meaning even if f belongs only to L1(〈0, T 〉;X) and u0 ∈ X. Thus,it can be used to further generalise the concept of the solution in a trivial way: a functionu ∈ C([0, T ];X) defined by (CS) is called a weak solution of (ACP) (some authors [26] use theterm mild solution). The weak solution is unique and satisfies the initial condition. Clearly, everyclassical solution is also a weak solution, while the converse is not true in general. The preciseinterpretation of the weak solution is given by the following theorem.



Theorem 10. [26, p. 108] Let f ∈ L1(〈0, T 〉;X), u0 ∈ X, and u is the weak solution of (ACP).Then u is the uniform limit on [0, T ] of the classical solutions un of

(ACP)n

u′n(t) +Aun(t) = fn

un(0) = u0n,

where fn ∈ C1([0, T ];X) and u0n ∈ D(A) are such that fn −→ f in L1(〈0, T 〉;X) and u0n −→ u0

in X.

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