Submitted to Mediterranean Journal of Mathematics

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1 Krešimir Burazin & Marko Erceg Submitted to Mediterranean Journal of Mathematics Abstract Inspired by the results of Ern Guermond and Caplain (007) on the abstract theory for Friedrichs symmetric positive systems we give the existence and uniqueness result for the initial (boundary) value problem for the non-stationary abstract Friedrichs system. Despite absence of well-posedness result for such systems there were already attempts for their numerical treatment by Burman Ern Fernandez (00) and Bui- Thanh Demkowicz Ghattas (03). We use the semigroup theory approach and prove that 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 examples such as the symmetric hyperbolic system the unsteady div-grad problem and the wave equation. Special attention was paid to the (generalized) unsteady Maxwell system. Keywords: symmetric positive first-order system semigroup abstract Cauchy problem Mathematics subject classification: 34Gxx 35F35 35F40 35F3 47A05 47D06 Department of Mathematics Department of Mathematics University of Osijek University of Zagreb Trg Ljudevita Gaja 6 Bijenička cesta 30 HR3000 Osijek Croatia HR0000 Zagreb Croatia kburazin@mathos.hr maerceg@math.hr This work is supported in part by the Croatian Science Foundation trough project Weak Convergence Methods ans Applications by the J.J. Strossmayer University of Osijek trough project Evolution Friedrichs systems and by DAAD 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 Centre for Applied Mathematics within the frame of the FP NUMERIWAVES project of the European Research Council. Corresponding author st February 06

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3 . Introduction An overview Symmetric positive systems of first-order linear partial differential equations were introduced by K. O. Friedrichs [] in 958 and today they are also known as Friedrichs systems. His immediate goal was the treatment of equations that change their type like the equations modelling the transonic fluid flow. The class of Friedrichs systems encompasses a wide variety of classical and neoclassical linear partial differential equations of continuum physics regardless of their type paired with various (initial) boundary conditions (Dirichlet Neumann Robin). This includes boundary value problems for some elliptic systems mixed initial and boundary value problems for parabolic and hyperbolic equations and boundary value problems for equations of mixed type such as the Tricomi equation. For some specific examples we refer to [ ]. The inclusion of such a variety of different problems into a single framework requires all different characteristics to be included as well. This naturally imposes a number of technical difficulties which many authors have since tried to surmount [7 8]. The setting of symmetric positive systems appeared to be convenient for the numerical treatment of various boundary value problems as well. Already Friedrichs considered the numerical solution of such systems by a finite difference scheme. The renewed interest in the theory of Friedrichs systems during the last decade resulted in development of new numerical schemes based on adaptations of the finite element method. Let us just mention [3] the Ph.D. thesis of Max Jensen [4] and most recently [ ]. As we understan some numerical algorithms based on mixed finite element methods are more suitable for first order systems then for higher order systems which makes the framework of Friedrichs systems more convenient for numerical treatment of some higher order equations. By applying some numerical scheme developed for Friedrichs systems to a particular equation of interest one often gets a numerical result (for that equation) that was not known before. Recently Ern Guermond and Caplain [6 9] suggested another approach to the Friedrichs systems which was as we understand inspired by their interest in the numerical treatment of Friedrichs systems. They expressed the theory in terms of operators acting in abstract Hilbert spaces and represented the boundary conditions in an intrinsic way thus avoiding the question of traces for functions from the graph space of the considered operator. Some open questions regarding the relationship of different representations of boundary conditions in the abstract setting were answered in []. The precise relationship between the classical Friedrichs theory and its abstract counterpart was investigated in [3 4] which resulted in new applications to second order equations which were also addressed in [5] and [9]. In the first reference the abstract theory was applied to the heat equation while in the second the homogenisation theory of the Friedrichs systems was developed and contrasted to the known results for stationary diffusion equation and the heat equation. Although some evolution (non-stationary) problems can be treated within the framework of abstract theory of Friedrichs systems developed in [9] by not making distinction between the time variable and space variables [5] the theory is not suitable for some standard problems like the initial-boundary value problem for non-stationary Maxwell system or the Cauchy problem for symmetric hyperbolic systems. In this paper we develop an abstract theory for non-stationary Friedrichs systems that can address these problems as well. More precisely we consider an abstract Cauchy problem in the Hilbert space involving a time independent abstract Friedrichs operator. In order to prove the well-posedness we use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille-Yosida generation theorem. Then a number of well-posedness results for different notions of a solution (classical/strong/weak) can be derived from the classical results for the abstract Cauchy problem [4 6]. The semigroup theory allows the treatment of semilinear problems [] as well resulting in the existence and uniqueness result for the semilinear Krešimir Burazin & Marko Erceg

4 non-stationary Friedrichs system. For some estimates on the solution of such problems we refer to [8]. Although the well-posedness result was not known there was already an aspiration for the numerical treatment of non-stationary Friedrichs systems [0 3]. Since a good well-posedness theorem is desirable for the convergence analysis of a numerical scheme we hope that our paper will pave the way for new numerical results in this direction. For example it would be interesting to see weather results of [] can be extended to Friedrichs systems as well. The paper is organised as follows: in the rest of this introductory section we recall the abstract setting of [9]. The main results are contained in the second section where we start by introducing the abstract non-stationary Friedrichs system then discuss some a priori bounds and finally prove that our operator satisfies conditions of the Hille-Yosida theorem. This enables us to give the existence and uniqueness results for different notions of solution. We also prove that the weak solution satisfies the starting equation in a certain vector valued L space and discuss the semilinear problem. In the third section we apply these result to a number of examples namely the Cauchy problem for the symmetric hyperbolic system the initial-boundary value problem for the unsteady Maxwell system the initial-boundary value problem for the unsteady div-grad problem and the initial-value problem for the wave equation. At the end in order to make the paper easier to read we have added an Appendix containing basic information regarding the application of the semigroup theory to the abstract Cauchy problem. Abstract Hilbert space formalism We start by describing the Hilbert space formalism introduced in [9] and recalling some basic results about the abstract Friedrichs systems. By L we denote a real Hilbert space identified with its dual L. Let D L be its dense subspace and L L : D L (unbounded) linear operators satisfying (T) ( ϕ ψ D) Lϕ ψ L = ϕ Lψ L (T) ( c > 0)( ϕ D) (L + L)ϕ L c ϕ L. These properties together will be refered as (T). By (T) 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) whose completion we denote by W 0. Analogously we could have defined L which by (T) leads to the norm that is equivalent to L. W 0 is continuously imbedded in L (as L is closable) with the image of W 0 equal to D( L) = D( L). Moreover when equipped with the graph norm these spaces are isometrically isomorphic. As L L : D L are continuous with respect to the graph topology on D they can be extended by density to a unique operator from L(W 0 ; L) (i.e. a continuous linear operator from W 0 to L). These extensions actually coincide with L and L (taking into account the isomorphism between W 0 and D( L)). For simplicity we shall drop the bar from notation and simply write L L L(W 0 ; L) prompted by the fact that (T) still hold for ϕ ψ W 0. Having in mind the Gelfand triple (the imbeddings are dense and continuous) the adjoint operator L L(L; W 0 ) defined by W 0 L L W 0 ( u L)( v W 0 ) W 0 L u v W0 = u Lv L Krešimir Burazin & Marko Erceg

5 satisfies L = L W0 as (T) implies W 0 L u v W0 = Lu v L = W 0 Lu v W0 u W 0. Therefore the continuous linear operator L : W 0 L W 0 from (W 0 L ) to W 0 has a unique continuous extension L to the whole L (the same holds for L and L instead of L and L ). In order to further simplify the notation we shall write L and L also for their extensions L and L respectively. By W := {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 W 0. Furthermore (W T ) is a Hilbert space [9 Lemma.]. In particular if we take L to be a partial differential operator (see an example below) than it 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 such that 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 [9 Lemma.3] that D is symmetric: ( u v W ) W Du v W = W Dv u W while ker D = W 0 [9 Lemma.4] justifies the usage of the term boundary operator. We are now ready to describe V : let V and Ṽ be two subspaces of W satisfying (V ) ( u V ) W Du u W 0 ( v Ṽ ) W Dv v W 0 (V ) V = D(Ṽ )0 Ṽ = D(V ) 0 where 0 stands for the annihilator. We shall refer to both (V) and (V) as (V). In practice one usually starts from a desired V uses the second equality in (V) as the definition of Ṽ and then it remains to check (V) and the first equality in (V). The following lemma is immediate. Lemma. If (T) and (V) hold then V and Ṽ are closed and ker D = W 0 V Ṽ. In order to get the well-posedness result (for the stationary case) one additional assumption that ensures the coercivity property of L and L is needed: (T3) ( µ 0 > 0)( ϕ D) (L + L)ϕ ϕ L µ 0 ϕ L. It is easy to see that the above property then holds for an arbitrary ϕ L. The operator L that satisfies (T) and (T3) for some L we call the abstract Friedrichs operator. The proof of the well-posedness result uses the following lemma. Lemma. ([9 Lemma 3.]) Under assumptions (T) (T3) and (V) the operators L and L are L coercive on V and Ṽ respectively; in other words: ( u V ) Lu u L µ 0 u L ( v Ṽ ) Lv v L µ 0 v L. Krešimir Burazin & Marko Erceg 3

6 Theorem. ([9 Theorem 3.]) If (T) (T3) and (V) hold then the restrictions of operators L V : V L and L Ṽ : Ṽ L are isomorphisms. The above theorem gives sufficient conditions on subspaces V and Ṽ that ensure wellposedness of the following problems: ) for given f L find u V such that Lu = f; ) for given f L find v Ṽ such that Lv = f. Its importance also arises from relative simplicity of geometric conditions (V) which in the case when L is a partial differential operator do not involve the notion of traces for functions in the graph space of L. Example. (Classical Friedrichs operator) Let d r N and Ω R d be an open and bounded set with a Lipschitz boundary Γ. Furthermore assume that the matrix functions A k W (Ω; M r (R)) k..d satisfy (F) A k is symmetric: A k = A k and C L (Ω; M r (R)). If we denote D := C c (Ω; R r ) L = L (Ω; R r ) and define operators L L : D L by formulæ Lu := d k (A k u) + Cu Lu := d k (A k u) + (C + d k A 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 (T) holds while (F) implies (T)). Therefore L and L can be uniquely extended to respective operators from L(L; W 0 ). Actually the formulæ for these extensions remain formally the same as for the starting operators with the distributional derivatives taken instead of the classical ones []. The graph space W in this example is given by W = { u L (Ω; R r ) : d } k (A k u) + Cu L (Ω; R r ) where we have to take distributional derivatives in the above formula. If we denote by ν = (ν ν... ν d ) L (Γ; R d ) the unit outward normal on Γ and define a matrix field on Γ by A ν := d ν k A k then for u v C c (R d ; R r ) the boundary operator D is given by W Du v W = A ν (x)u Γ (x) v Γ (x)ds(x). If we additionally assume (F) ( µ 0 > 0) C + C + Γ d k A k µ 0 I (a.e. on Ω) then the property (T3) is also satisfied. The inequality above is meant in the sense of the order on symmetric matrices. Krešimir Burazin & Marko Erceg 4

7 Remark. If we start in the previous example from the beginning with the coefficients defined on the whole R d instead of Ω and consider the operators L L : D L defined by above formulæ with D := C c (R d ; R r ) L = L (R d ; R r ) then one can easily see that W = W 0 = ker D [ 7 4] which implies that the boundary operator D is trivial. Therefore we can take V = Ṽ = W and these spaces satisfy conditions (V) which alows us to consider the initial value problem without any other constrains in the non-stationary case.. Non-stationary problem The abstract Cauchy problem In the rest of the paper we shall consider an initial-boundary value problem for a nonstationary Friedrichs system. To be specific using the notation from the previous section we shall consider the abstract Cauchy problem: (P) { u (t) + Lu(t) = f u(0) = u 0 where u : [0 T L T > 0 is the unknown function while the right-hand side f : 0 T L the initial data u 0 L and the abstract Friedrichs operator L are given. More precisely we assume that L does not depend on the time variable t and satisfies (T) so that it can be extended to the continuous linear operator L W 0 as in the introductory section. Instead of the positivity assumption (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 the case 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 (F) can be weakened to (F ) C + C + d k A k 0 (a.e. on Ω) as this is enough to ensure the validity of (T3 ). Actually if L satisfies only (F) the positivity condition (F) can be achieved by substituting v := e λt u for some suitable λ > 0: then the corresponding non-stationary system becomes t v + (L + λi)v = e λt f where I is the identity matrix and since all matrices appearing in the above expression are bounded we can choose λ large enough so that (F) is satisfied. Therefore the operator L + λi and its formal adjoint satisfy (T) (T3). Note also that the initial condition u(0 ) = u 0 is equivalent to v(0 ) = u 0 while the graph space of operator L + λi is the same as the graph space of L with equivalent norms. In order to fully describe problem (P) (see Appendix) it remains to be clarified what is the domain D(L) of L. As the equation (P) should hold in L we can take any subspace of the graph space W. If L is a partial differential operator then the choice of domain of L corresponds to given boundary conditions and it is then natural to take for D(L) to be some subspace V satisfying (V). Then one can easily prove a similar result as in Lemma if the weaker assumption (T3 ) is taken instead of (T3). Krešimir Burazin & Marko Erceg 5

8 Lemma 3. Under assumptions (T) (T3 ) and (V) the operators L and L are L-accretive on V and Ṽ respectively i.e. they satisfy ( u V ) Lu u L 0 ( v Ṽ ) Lv v L 0. If u is the classical solution of (P) on [0 T (see the Appendix for definitions of different notions of a solution) one can easily derive an a priori estimate ( t [0 T ]) u(t) L e t ( u 0 L + t 0 ) f(s) L from which the uniqueness of classical solution is immediate. Indeed from (P) we have and by (T3 ) it follows u (t) u(t) L + Lu(t) u(t) L = f(t) u(t) L d dt u(t) L f(t) u(t) L u(t) L + f(t) L from where after applying the differential form of the Gronwall inequality we get the a priori estimate. Here we have assumed that the right-hand side of (P) is square-integrable i.e. f L ( 0 T ; L). Clearly this is not sufficient for the existence of the classical solution and precise sufficient 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 the Appendix let us take a subspace V of W satisfying (V) and define an operator A : V L by A := L V. We can now rewrite our problem (P) in terms of operator A: (P ) { u (t) Au(t) = f u(0) = u 0 and since the domain V of A is equipped with the topology inherited from L (and not the graph norm) A is unbounded in general. Our goal is to show that A is an infinitesimal generator of a strongly continuous semigroup (C 0 -semigroup) on L. If we succeed then our problem (P ) (and thus also (P)) can be fitted in the setting of abstract Cauchy problems of the semigroup theory. The following theorem is the main result of this paper. Theorem. A is the infinitesimal generator of a contraction C 0 -semigroup (T (t)) t 0 on L. Dem. According to the Hille-Yosida generation theorem (Theorem 7) it is necessary and sufficient to show that a) V is dense in L b) A is closed c) ρ(a) 0 and R λ (A) λ for every λ > 0. Density of V is a direct consequence of Lemma. In order to prove (b) let us take a sequence (u n ) in V such that u n u and Au n f both in L. We need to show that u V and f = Au. It is easy to show that (u n ) is a Cauchy sequence in the complete graph space W. Indeed u n u m W = u n u m L + Lu n Lu m L = u n u m L + Au n Au m L Krešimir Burazin & Marko Erceg 6

9 and the claim follows by the assumption that (u n ) and (Au n ) are convergent (therefore Cauchy) sequences in L. Thus (u n ) is convergent in W and because V is closed in W we additionally have that the limit v is in V. The convergence in the graph norm gives us u n v and Au n Av in L which implies u = v V and f = Av = Au. In order to prove the last statement let us define L λ := λi + L and a corresponding L λ := λi + L for an arbitrary λ > 0 and let us show that L λ is an isomorphism from V to L. The idea of proof is to show that L λ and L λ satisfy (T) (T3) and then apply Theorem. The condition (T) is trivially satisfied (T) is satisfied if we take c + λ as the constant and since L and L satisfy (T3 ) we have (L λ + L λ )u u L = (L + L)u u L + λ u u L λ u L for every u L which is exactly (T3). Before applying Theorem it is important to notice that the corresponding graph space W λ and the boundary operator D λ of operator L λ do not depend on λ. Indeed one can verify that for every λ > 0 we have W λ = W and D λ = D which implies that subspace V satisfy conditions (V) in the graph space of operator L λ. Now we have that L λ V : V L is an isomorphism and therefore ρ(a) 0. Finally Lemma gives us L λ V λ = R λ (A) λ because R λ (A) = (L λ V ). Q.E.D. Remark. Instead of using the Hille-Yosida theorem in the proof of the previous theorem we could alternatively (like in [4 p. 4] and [6 p. 4]) 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 the Lumer-Phillips theorem. After we have established that A generates a contraction C 0 -semigroup we can use some classical results (see Appendix for some of them and [ 6] for a more extensive presentation) to derive various conclusions about solvability of (P). In the following corollary we shall summarise results that follow from the results stated in the Appendix. Corollary. Let L be an operator that satisfies (T) and (T3 ) V a subspace of its graph space satisfying (V) and f L ( 0 T ; L). a) Then for every u 0 L the problem (P) has the unique weak solution u C([0 T ]; L) given by u(t) = T (t)u 0 + t 0 T (t s)f(s)ds t [0 T ] where (T (t)) t 0 is as in Theorem. ( ) b) If additionally u 0 V and f W ( 0 T ; L) C([0 T ]; L) L ( 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 u 0 L + t 0 f(s) L ds. By the definition of weak solution it is clear that it satisfies the initial condition while the equation is satisfied in the meaning of Theorem 0. However in our setting we can show that the weak solution satisfies the equation in a certain space (a similar observation has been done in [4 p. 406]). Krešimir Burazin & Marko Erceg 7

10 Theorem 3. Let u 0 L f L ( 0 T ; L) and let u be the weak solution of (P). Then u Lu f L ( 0 T ; W 0 ) and u + Lu = f in L ( 0 T ; W 0 ). Dem. According to Theorem 0 there exists a sequence of the classical solutions u n to (P ) with right hand sides f n C ([0 T ]; L) and initial conditions u 0n V such that f n f in L ( 0 T ; L) u 0n u 0 in L and u n u in C([0 T ]; L). Since L L(L; W 0 ) we have Lu n Lu in C([0 T ]; W 0 ) which implies the convergence in the space L ( 0 T ; W 0 ). As L is continuously embedded in W 0 we also have f n f in L ( 0 T ; W 0 ). Thus u n = Lu n + f n Lu + f in L ( 0 T ; W 0 ). On the other hand u n u in the space of vector valued distributions D ( 0 T ; W 0 ) which implies u n u in D ( 0 T ; W 0 ) (see [4 p. 8]). The uniqueness of the limit in the space of vector valued distributions gives us u + Lu = f in D ( 0 T ; W 0 ) and since Lu f L ( 0 T ; W 0 ) and L ( 0 T ; W 0 ) is continuously embedded in D ( 0 T ; W 0 ) we also have u L ( 0 T ; W 0 ). Q.E.D. Remark. As it is well known semilinear problems can also be treated via the semigroup theory [ 6]. Therefore we can get the existence and uniqueness result for the abstract Cauchy problem { u (t) + Lu(t) = f(t u(t)) u(0) = u 0 where L is an abstract Friedrichs operator and the right-hand side f : 0 T L L depends also on the unknown function. Usual assumptions on f that ensure existence and uniqueness of the weak solution are continuity and some kind of Lipschitz continuity in the last variable (see [ Ch. 4] and [6 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-stationary Friedrichs systems and apply the results of the previous section. Initial problem for symmetric hyperbolic system We consider the initial problem for first-order system d t u + k (A k u) + Cu = f in 0 T R d (HS) u(0 ) = u 0 where T > 0 is some given time and the coefficients A k W (R d ; M d (R)) k..n C L (R d ; M d (R)) do not depend on time. For the right-hand side and the initial datum we suppose f L ( 0 T ; L (R d ; R d )) and u 0 L (R d ; R d ) while u : [0 T R d R d is the unknown function. We also suppose that each A k is symmetric a.e. on R d One can easily see that (HS) fits in our framework of non-stationary Friedrichs systems with the operator L taken to be d Lu := k (A k u) + Cu. Krešimir Burazin & Marko Erceg 8

11 Indeed the symmetry condition (F) is trivially satisfied while the positivity condition (F) can be obtained by using the substitution v := e λt u for some λ > 0 large enough as remarked before. Then our Friedrichs system reads t v + (L + λi)v = e λt f the initial condition remains v(0 ) = u 0 and the graph space of the operator L + λi is the same as the graph space of L with equivalent norms: W = {w L (R d ; R d ) : Lw L (R d ; R d )}. Now as a consequence of the remark closing the introductory section and Corollary we have the result of existence and uniqueness of the solution to (HS). ( ) Theorem 4. Let f W ( 0 T ; L (R d ; R d )) C([0 T ]; L (R d ; R d )) L ( 0 T ; W ) and u 0 W. Then the abstract initial-value problem u + has the unique classical solution given by u(t) = e λt T(t)u 0 + d k (A k u) + Cu = f t 0 u(0) = u 0 e λ(t s) T(t s)f(s)ds t [0 T ] where (T(t)) t 0 is the contraction C 0 -semigroup generated by L λi. Remark. The above formula could have been written in terms of the C 0 -semigroup S(t) := e λt T(t) with the infinitesimal generator L [6 p. ]. Remark. A similar existence and uniqueness result for the Cauchy problem for the symmetric hyperbolic systems under some additional assumptions on coefficients can be found in [0]. Time-dependent Maxwell system Let Ω R 3 be open and bounded with a Lipschitz boundary Γ T > 0 Σ ij L (Ω; M 3 (R)) i j { } f f L ( 0 T ; L (Ω; R 3 )) µ ε W (Ω; M 3 (R)) and constants µ 0 ε 0 R + such that for every x Ω µ(x) and ε(x) are symmetric and µ µ 0 I ε ε 0 I. We consider the generalised time-dependent Maxwell system: (MS) { µ t H + rot E + Σ H + Σ E = f ε t E rot H + Σ H + Σ E = f in 0 T Ω where E H : [0 T Ω R 3 are the unknown functions. Remark. If we take f 0 f = J Σ = Σ = Σ 0 we will get the (standard) Maxwell system in a linear nonhomogeneous anisotropic medium. In that case H E ε Σ µ J represent the magnetic and electric fields the electric permeability the conductivity of the medium the magnetic susceptibility of the material in Ω and the applied current density respectively. The principal square roots µ(x) and ε(x) 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 µ (x) := µ(x) ε (x) := ε(x). Moreover we have µ ε W (Ω; M 3 (R)) both µ (x) and ε (x) are symmetric for every x Ω with µ µ 0 I ε ε 0 I. The last property implies that µ (x) and ε (x) are regular for every x and moreover µ ε W (Ω; M 3 (R)) where µ (x) := µ (x) and ε (x) := ε (x). Krešimir Burazin & Marko Erceg 9

12 After introducing the substitutions (using the previous notation) u := [ u u ] [ ] µ H = ε E and multiplying the first three equations by µ and the last three by ε we can write our system as t u + Lu = F where where Lu := [ µ rot (ε ] [ u ) µ Σ ε rot (µ + µ µ Σ ε u ) ε Σ µ ε Σ ε ] [ ] [ ] u µ f F = u ε. f To conclude that L is a classical Friedrichs operator we still need to do some computations: µ rot (ε u ) = µ = 3 k (B k ε u ) 3 ( 3 ) k (µ Bk ε u ) ( k µ )Bk ε u B = B = B 3 = are matrices corresponding to the differential operator rot. We can analougously rewrite the last three components ε rot (µ u ) and get: [ µ rot (ε ] u ) ε rot (µ = u ) 3 ([ 0 µ B k ε k ε Bk µ 0 ( 3 [ + ] ) u 0 ( k µ )Bk ε ( k ε )B k µ 0 ] ) u. If we define [ 0 µ A k := B k ε ] ε Bk µ k [ 0 ( C := k µ )Bk ε ] [ µ Σ ( k ε )B k µ + µ µ Σ ε ] 0 ε Σ µ ε Σ ε then one can easily see that L takes the form Lu = 3 k (A k u) + Cu of the classical Friedrichs partial differential operator. Indeed the symmetry condition (F) is trivially satisfied while the positivity condition (F) can be obtained using the substitution v := e λt u as in the previous example. Therefore without loss of generality let us assume that our operator L satisfies the positivity condition (F). Krešimir Burazin & Marko Erceg 0

13 Remark. As the system above is a symmetric hyperbolic system the Cauchy problem on whole R 3 is contained in the setting of the previous example so the existence and uniqueness result is immediate. Using that we can get the existence and uniqueness result for the Cauchy problem of the starting Maxwell system (MS). Obviously the spaces involved are L = L (Ω; R 3 ) L (Ω; R 3 ) { [ ] [ ] } u rot (µ u W = u = L : ) u rot (ε L u ) { [ ] [ ] } u µ u = u = L : u ε L rot(ω; R 3 ) L rot(ω; R 3 ) u where L rot(ω; R 3 ) stands for the graph space of differential operator rot. Because µ and ε are uniformely bounded from below and above the graph norm L is equivalent with the norm [ ] µ u u := ε u L rot (Ω;R 3 ) L rot (Ω;R3 ) on W. If we denote by ν = (ν ν ν 3 ) L (Γ; R 3 ) the unit outward normal on Γ matrix field A ν can be written as a block matrix where [ A ν = 0 µ A rot ν ε ε A rot ν µ 0 A rot ν = 0 ν 3 ν ν 3 0 ν ν ν 0 is a matrix corresponding to the operator rot. We can describe the boundary operator D in terms of the trace operator T H : H (Ω; R 3 ) H (Γ; R 3 ) and the tangential trace operator (see [] [7]) T rot : L rot(ω; R 3 ) H (Γ; R 3 ). Indeed for u v C c (R 3 ; R 6 ) we have W Du v W = A ν (x)u Γ (x) v Γ (x)ds(x) Γ ] = H µ Trot (ε u ) T H v H H ε Trot (µ u ) T H v H = H µ Trot (ε v ) T H u H H ε Trot (µ v ) T H u H where the second equation can be extended by density to (u v) W H (Ω; R 6 ) and the third one to (u v) H (Ω; R 6 ) W. Also note that in the above expression we have multiplication of a functional from H (Γ; R 3 ) by matrix Lipschitz functions µ and ε which is well-defined as follows H µ g z H := H g µ z Γ H z H (Γ; R 3 ) g H (Γ; R 3 ) and analogously for ε. 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 trace operators). Now we take V = Ṽ = {u W : T rot(ε u ) = 0} = {u W : T rot E = 0} Krešimir Burazin & Marko Erceg

14 and verify the conditions (V) and (V). Note that u V if and only if [H E] L rot(ω; R 3 ) L rot0 (Ω; R3 ) where L rot0 (Ω; R3 ) := Cl L rot (Ω;R 3 ) C c (Ω; R 3 ). To prove (V) we need to show ( u V ) W Du u W = 0. Since u = [u u ] V implies µ u L rot(ω; R 3 ) and ε u L rot0 (Ω; R3 ) there is a sequence (u n ) = ([u n un ] ) in H (Ω; R 3 ) C c (Ω; R 3 ) such that u n u in W. As T H u n = 0 and T rot (ε u n ) = ν T H (ε u n ) = 0 using computed identity for D and its continuity we get W Du u W = lim n W Dun u n W = lim n 0 = 0 which proves (V). In order to prove (V) let us first take arbitrary u v V and the corresponding approximating sequences (u n ) (v n ) from H (Ω; R 3 ) C c (Ω; R 3 ) as in the previous case. Similarly like before we have W Dv u W = lim n W Dvn u n W = lim n 0 = 0 as T rot (ε u n ) = T H vn = 0 which gives us V D(V )0. For the opposite inclusion we need to show that for an arbitrary u = [u u ] W condition ( v V ) W Dv u W = 0 implies T rot (ε u ) = 0. If we put in the above equality v = [v 0] v H (Ω; R 3 ) we get W Dv u W = H µ Trot (ε u ) T H v H = 0. Since v H (Ω; R 3 ) is arbitrary T H is surjective and the mapping z µ z is from H (Γ; R 3 ) onto H (Γ; R 3 ) we get T rot (ε u ) = 0 and (V) is obtained. Let us summarise using Corollary all that has been shown in the following theorem. Theorem 5. Let E 0 L rot0 (Ω; R3 ) H 0 L rot(ω; R 3 ) and let f f C([0 T ]; L (Ω; R 3 )) satisfy at least one of the following two conditions i) f f W ( 0 T ; L (Ω; R 3 )); ii) µ f L ( 0 T ; L rot(ω; R 3 )) ε f L ( 0 T ; L rot0 (Ω; R3 )). Then the abstract initial-boundary value problem µh + rot E + Σ H + Σ E = f has the unique classical solution given by [ ] [ ] [ H µ 0 µ (t) = E 0 ε T(t) ε E 0 εe rot H + Σ H + Σ E = f E(0) = E 0 H(0) = H 0 ν E Γ = 0 H 0 ] [ ] µ 0 t + 0 ε T(t s) 0 where (T(t)) t 0 is the contraction C 0 -semigroup generated by L. [ ] µ f (s) ε ds t [0 T ] f (s) Remark. The above theorem is valid if the corresponding Friedrichs operator satisfies (F ). As written before if this is not the case then the positivity condition can be obtained by a simple change of variable. The statement of the above theorem should also be changed accordingly. Krešimir Burazin & Marko Erceg

15 We can get a similar result if we take the boundary condition ν H Γ ν E Γ = 0. More precisely it can be shown that spaces = 0 instead of V = Ṽ = {u W : T rot(µ u ) = 0} = {u W : T rot H = 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 [] [4]. Unsteady div-grad problem Let Ω R d be open and bounded with a Lipschitz boundary Γ f L ( 0 T ; L (Ω; R d )) f L ( 0 T ; L (Ω; R)) and c > 0. We consider a system of equations (DG) t q + p = f c in 0 T Ω tp + div q = f with the unknowns p : [0 T Ω R and q : [0 T Ω R d. Remark. These are linearised equations for the propagation of sound in the inviscid elastic and compressible fuid describing small disturbances [5]. Here the unknowns q and p correspond to the velocity of the fluid and the pressure while the constant c stands for the speed of sound in the fluid. After introducing the substitution we get an evolution Friedrichs system u := [ u u ] = [ c p ] q t u + Lu = F [ ] [ ] cf c div u where F = and Lu := is the classical Friedrichs operator satisfying (F) and f c u (F ) with C = 0 and A k = c e e k+ + c e k+ e M +d (R). Obviously we have W = H (Ω) L div (Ω; Rd ) W 0 = H 0(Ω) L div0 (Ω; Rd ) = Cl W C c (Ω; R +d ) where L div (Ω; Rd ) is the graph space of differential operator div and L div0 (Ω; Rd ) is the closure of C c (Ω; R d ) in the graph norm of L div (Ω; Rd ). Remark. As the system above is a symmetric hyperbolic system the Cauchy problem on whole R d is contained in the setting of the first example so the existence and uniqueness result is immediate which also implies the existence and uniqueness result for the Cauchy problem of the unsteady div-grad problem. In this case we have V = Ṽ = W = H (R d ) L div (Rd ; R d ). The matrix field A ν can be written as a block matrix [ ] 0 cν A ν = cν 0 Krešimir Burazin & Marko Erceg 3

16 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 T H : H (Ω) H (Γ) and the normal trace operator (see [] [7]) T div : L div (Ω; Rd ) H (Γ): W Du v W = A ν (x)u Γ (x) v Γ (x)ds(x) Γ = c H T divu T H v H + c H T divv T H u H for u v C c (Ω; R +d ). The second equality is actually valid for all (u v) W W by the density argument. We shall study two possible boundary conditions the first one is given by V = Ṽ = {u W : T H u = 0} = {u W : T H p = 0} = H 0(Ω) L div (Ω; Rd ) and it clearly corresponds to the boundary condition p Γ = 0. For the subspace V we can check the condition (V) analogously as in the previous example as well as V D(V ) 0. For the proof of the opposite inclusion in (V) we need to show that for an arbitrary u W the condition ( v V ) W Dv u W = 0 implies T H u = 0. If we put in the above equality v = [0 v ] v L div (Ω; Rd ) we get W Dv u W = c H T divv T H u H = 0 which implies T H u = 0 by the arbitrariness of v L div (Ω; Rd ) and the surjectivity of T div. Using Corollary we can summarise the above computations in the following theorem. Theorem 6. Let q 0 L div (Ω; Rd ) p 0 H 0 (Ω) and let functions f C([0 T ]; L (Ω; R d )) and f C([0 T ]; L (Ω)) satisfy at least one of the following two conditions i) f W ( 0 T ; L (Ω; R d )) f W ( 0 T ; L (Ω)); ii) f L ( 0 T ; L div (Ω; Rd )) f L ( 0 T ; H 0 (Ω)). Then the abstract initial-boundary-value problem q + p = f c p + div q = f q(0) = q 0 p(0) = p 0 p Γ = 0 has the unique classical solution given by [ ] [ ] [ p c 0 = T(t) c p ] [ ] 0 c 0 t [ cf + T(t s) q 0 I 0 I where (T(t)) t 0 is the contraction C 0 -semigroup generated by L. q 0 0 f ] ds t [0 T ] We can get a similar result for the another possible boundary condition ν q = 0 Γ i.e. for the subspaces V = Ṽ = {u W : T divu = 0} = {u W : T div q = 0} = H (Ω) L div0 (Ω; Rd ). Krešimir Burazin & Marko Erceg 4

17 Initial-value problem for the wave equation Let us now consider the initial-value problem tt u c u = f in 0 T R d (WE) u(0 ) = u 0 t u(0 ) = u 0 where T > 0 is a given constant c > 0 represents the propagation speed of the wave f L ( 0 T ; L (R d )) is the external force u 0 H (R d ) and u 0 L (R d ) are the inital position and velocity respectively while u : [0 T R d R is the unknown function which represents the displacement. After a change of variable [ ] [ ] u t u u := = c u u we get the same system t u + Lu = F as in the previous example with f 0 and f = c f. Note that the last d equations of this system are actually the Schwarz symmetry relations while the first one results from the wave equation we started with. The initial condition transforms to u(0) = [ u (0) u (0) ] [ = u 0 c u 0 ] =: u 0. 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 = Ṽ = W = H (R d ) L div (Rd ; R d ) we have the existence result. However due to our change of variable we only have information about the derivatives of the original unknown. Therefore in order to find u one must solve the problem { u (t) = u (t) (odep). u(0) = u 0 Let us illustrate this ( in the case of the classical solution; ) if we additionally assume f W ( 0 T ; L (R d )) C([0 T ]; L (R d )) L ( 0 T ; H (R d )) u 0 H (R d ) and u 0 L (R d ) then (by Corollary ) the abstract Cauchy problem { u (t) + Lu(t) = F (P) u(0) = u 0 has the classical solution u C ( 0 T ; L (R d ; R d+ )) C([0 T ]; L (R d ; R d+ )) C( 0 T ; W ). Therefore u C ( 0 T ; L (R d )) C([0 T ]; L (R d )) C( 0 T ; H (R d )) u C ( 0 T ; L (R d ; R d )) C([0 T ]; L (R d ; R d )) C( 0 T ; L div (Rd ; R d )) which ensures that the problem (odep) has the unique solution u belonging to C ( 0 T ; L (R d )) C ( 0 T ; H (R d )) C([0 T ]; H (R d )). Note that this u already satisfies the first initial condition in (WE). Let us show that it also satisfies the wave equation in C( 0 T ; L (R d )). Since : H (R d ) L (R d ) is a continuous linear operator it follows that the operator (which we denote the same) defined by ( u)(t) := (u(t)) is a linear continuous operator C ( 0 T ; H (R d )) C ( 0 T ; L (R d )) and C( 0 T ; H (R d )) C( 0 T ; L (R d )) which commutes with the time derivative: (u ) = ( u). Using this the last d equations in (P) and (odep) we get u = ( c u) in the space C( 0 T ; L (R d ; R d )) which together with the initial conditions for u and u imply u = c u. Note that as an additional consequence we have u C( 0 T ; L div (Rd )). Substituting u and u in the first equation in (P) we get (in C( 0 T ; L (R d ))) u = u = c div u + f = c div ( c u) + f = c u + f. It remains to prove that u satisfies the second initial condition in (WE) which easily follows from u = u in C( 0 T ; L (R d )) u C([0 T ]; L (R d )) and u (0) = u 0. Krešimir Burazin & Marko Erceg 5

18 Remark. A similar existence and uniqueness result for the initial-value problem for the wave equation within the framework of a semigroup theory can be found in [6]. Remark. Using results of the unsteady div-grad problem we can get the existence and uniqueness results for the initial-boundary value problem for the wave equation with two possible boundary conditions: t u 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 have used in the paper (details can be found in [] [6] and [4 Ch. XVII]). Throughout the section (X X ) will denote a real Banach space and the norm on the space of bounded linear operators on X L(X). A family (T (t)) t 0 of bounded linear operators on X is called strongly continuous semigroup (C 0 -semigroup) if it satisfies i) T (0) = I (identity) ii) ( t s 0) T (t + s) = T (t)t (s) iii) ( x X) t T (t)x is continuous. As a consequence of the uniform boundedness theorem one can show [6 p. 4] that for every C 0 -semigroup (T (t)) t 0 there exist constants ω 0 and M such that ( t 0) T (t) Me ωt. If in the above expresion we can take ω = 0 and M = then (T (t)) t 0 is called a contraction C 0 -semigroup. The infinitesimal generator of a C 0 -semigroup (T (t)) t 0 is a map A : D(A) X defined by Ax := lim t 0 t ( T (t)x x on in its domain D(A) consisting of those x X for which the above limit exists. It can be shown 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 C 0 -semigroup and if it is whether we can find an explicit formula for this semigroup. For the latter question in general situations there are no computable formulæ but for the first one the answer is given by the famous Hille-Yosida theorem. Theorem 7. (Hille-Yosida) [6 p. 8] A linear operator A : D(A) X X is the infinitesimal generator of some contraction C 0 -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 R λ (A) λ where ρ(a) := {λ R : λi A : D(A) X is isomorphism} is the resolvent set of A while is the resolvent of A. ) R λ (A) := (λi A) λ ρ(a) Krešimir Burazin & Marko Erceg 6

19 The abstract Cauchy problem One of the most prominent applications of the semigroup theory is in solving the abstract Cauchy problem (ACP) { u (t) + Au(t) = f u(0) = u 0 where a linear operator A : D(A) X X u 0 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 C ( 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) 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) [6 p. 06] 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 of the Hille-Yosida theorem in obtaining the existence and uniqueness result for (ACP). Theorem 8. [6 p. 0] Let A be a densely defined operator with a nonempty resolvent set ρ(a) and let f 0. The problem (ACP) has for every u 0 D(A) the unique classical solution u on [0 if and only if A is the infinitesimal generator of a C 0 -semigroup (T (t)) t 0. Moreover u(t) = T (t)u 0 for every t 0. If the semigroup from the previous theorem is additionally a contraction C 0 -semigroup then from the formula for the solution we get the estimate ( t 0) u(t) X u 0 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 C 0 -semigroup (T (t)) t 0. If f C([0 T ]; X) and u 0 D(A) then one can easily see that any classical solution u of (ACP) satisfies (CS) u(t) = T (t)u 0 + t 0 T (t s)f(s)ds t [0 T ] which proves the uniqueness of classical solution. If (T (t)) t 0 is a contraction C 0 -semigroup then we also have the estimate ( t [0 T ]) u(t) X u 0 X + t 0 f(s) X ds. However in order to check that the right-hand side of (CS) really defines a solution one needs to verify that the last term above is (continuously) differentiable [6 p. 07] which requires some additional assumptions on f. Theorem 9. [ p. 5] Let f C([0 T ]; X) u 0 D(A) and assume that at least one of the following conditions is satisfied: i) f W ( 0 T ; X); ii) f L ( 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 L ( 0 T ; X) and u 0 X. Thus it can be used to further generalise the concept of the solution in a trivial way: a function u C([0 T ]; X) defined by (CS) is called a weak solution of (ACP) (some authors [6] use the term mild solution). The weak solution is unique and satisfies the initial condition. Clearly every classical solution is also a weak solution while the converse is not true in general. The precise interpretation of the weak solution is given by the following theorem. Krešimir Burazin & Marko Erceg 7

20 Theorem 0. [6 p. 08] Let f L ( 0 T ; X) u 0 X and u is the weak solution of (ACP). Then u is the uniform limit on [0 T ] of the classical solutions u n of (ACP) n { u n (t) + Au n (t) = f n u n (0) = u 0n where f n C ([0 T ]; X) and u 0n D(A) are such that f n f in L ( 0 T ; X) and u 0n u 0 in X. References [] Nenad Antonić Krešimir Burazin: Graph spaces of first-order linear partial differential operators Math. Communications 4() (009) [] Nenad Antonić Krešimir Burazin: Intrinsic boundary conditions for Friedrichs systems Comm. Partial Diff. Eq. 35 (00) [3] Nenad Antonić Krešimir Burazin: Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems J. Diff. Eq. 50 (0) [4] Nenad Antonić Krešimir Burazin Marko Vrdoljak: Second-order equations as Friedrichs systems Nonlinear analysis: RWA 4 (04) ; [5] Nenad Antonić Krešimir Burazin Marko Vrdoljak: Heat equation as a Friedrichs system Journal of Mathematical Analysis and Applications 404 (03) [6] Tan Bui-Thanh Leszek Demkowicz Omar Ghattas: A unified discontinuous Petrov-Galerkin method and its analysis for Friedrichs systems SIAM J. Numer. Anal. 5 (03) [7] Krešimir Burazin: Contributions to the theory of Friedrichs and hyperbolic systems (in Croatian) Ph.D. thesis University of Zagreb kburazin/papers/teza.pdf [8] Krešimir Burazin Marko Erceg: Estimates on the weak solution of abstract semilinear Cauchy problems Electronic Journal of Differential Equations 04/94 (04) -0 [9] Krešimir Burazin Marko Vrdoljak: Homogenisation theory for Friedrichs systems Communications on Pure and Applied Analysis 3/3 (04) [0] Erik Burman Alexandre Ern Miguel A. Fernandez: Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems SIAM J. Numer. Anal. 48 (00) [] Thierry Cazenave Alain Haraux: An Introduction to Semilinear Evolution Equations Oxford University Press 998. [] Michel Cessenat: Mathematical methods in electromagnetism World Scientific 996. [3] Daniele Antonio Di Pietro Alexandre Ern: Mathematical aspects of discontinuous Galerkin methods Springer 0. [4] Robert Dautray Jacques-Louis Lions: Mathematical analysis and numerical methods for science and technology Vol.V Springer 99. [5] Alexandre Ern Jean-Luc Guermond: Theory and practice of finite elements Springer 004. [6] Alexandre Ern Jean-Luc Guermond: Discontinuous Galerkin methods for Friedrichs systems. I. General theory SIAM J. Numer. Anal. 44 (006) [7] Alexandre Ern Jean-Luc Guermond: Discontinuous Galerkin methods for Friedrichs systems. II. Second-order elliptic PDEs SIAM J. Numer. Anal. 44 (006) [8] Alexandre Ern Jean-Luc Guermond: Discontinuous Galerkin methods for Friedrichs systems. III. Multifield theories with partial coercivity SIAM J. Numer. Anal. 46 (008) [9] Alexandre Ern Jean-Luc Guermond Gilbert Caplain: An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs systems Comm. Partial Diff. Eq. 3 (007) [0] Hector O. Fattorini: The Cauchy problem Cambridge University Press 983 [] Kurt O. Friedrichs: Symmetric positive linear differential equations Comm. Pure Appl. Math. (958) [] Marlis Hochbruck Tomislav Pažur Andreas Schulz Ekkachai Thawinan Christian Wieners: Efficient time integration for discontinuous Galerkin approximations of linear wave equations Technical report Karlsruhe Institute of Technology 03 [3] Paul Houston John A. Mackenzie Endre Süli Gerald Warnecke: A posteriori error analysis for numerical approximation of Friedrichs systems Numer. Math. 8 (999) Krešimir Burazin & Marko Erceg 8

21 [4] Max Jensen: Discontinuous Galerkin methods for Friedrichs systems with irregular solutions Ph.D. thesis University of Oxford dma0mpj/thesisjensen.pdf [5] Lev D. Landau Evgenĭ M. Lifshitz: Fluid Mechanics Pergamon Press 987. [6] Amnon Pazy: Semigroups of Linear Operator and Applications to Partial Differential Equations Springer 983. [7] Ralph S. Phillips Leonard Sarason: Singular symmetric positive first order differential operators Journal of Mathematics and Mechanics 5 (966) [8] Jeffrey Rauch: Symmetric positive systems with boundary characteristic of constant multiplicity Trans. Amer. Math. Soc. 9 (985) Krešimir Burazin & Marko Erceg 9