On Maximal Regularity for a Class of Evolutionary Equations.

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1 On Maximal Regularity for a Class of Evolutionary Equations. arxiv: v1 [math.ap] 4 Apr 216 Rainer Picard, Sascha Trostorff Institut für Analysis, Fachrichtung Mathematik, Technische Universität Dresden Marcus Waurick Department of Mathematical Sciences, University of Bath rainer.picard@tu-dresden.de, sascha.trostorff@tu-dresden.de, marcus.waurick@tu-dresden.de The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to establishing that the operator sum considered with its natural domain is already closed. For this we use structural constraints of the coefficients rather than semi-group strategies or sesqui-linear form methods, which would be difficult to come by for our general problem class. Our approach, although limited to the Hilbert space case, complements known strategies for approaching maximal regularity and extends them in a different direction. The abstract findings are illustrated by re-considering some known maximal regularity results within the framework presented. Keywords and phrases: maximal regularity, evolutionary equations, material laws, coupled systems Mathematics subject classification 21: 35B65, 35D35, 58D25 Contents Introduction 2 1 A brief description of the framework of evolutionary equations 3 2 The main result 5 1

2 3 Some Examples The heat equation A second order equation A second order integro-differential equation A partial differential equation of fractional type Introduction The issue of maximal regularity has received much attention as an important property of certain partial differential equations and more abstractly as a feature of a class of evolution equations. In a Hilbert space setting, the typical situation thus refers to an abstract operator equation in L 2,loc [, [,H of the form u Au = f, 1 for some given f L 2,loc [, [,H, H a Hilbert space. Moreover, u: ], [ H is a measurable function withu being its weak derivative andais the abstract linear operator on L 2,loc [, [,H induced by an operator A, assumed to be the infinitesimal generator of a one-parameter C -semi-group, i.e. Aut := Aut. If we solve equation 1 for u subject to homogeneous initial conditions, we can expect u to be at best only continuous. Thus, u is a so-called mild solution of 1, that is, u solves the equation in question in an integrated form. To obtain better regularity behaviour one is interested in the case, where for any given f, the corresponding solution u is such that u and Au both belong to L 2,loc [, [,H and, hence, u literally solves 1 in L 2,loc [, [,H. This property is commonly attributed to the semi-group generator A and one says in this case that A admits maximal L 2 -regularity. A standard situation is that A is a non-negative selfadjoint operator and so, if A = C C for some closed and densely defined linear operator C, the corresponding evolution equation admits maximal regularity as can be easily seen in this simple case with the help of the spectral theorem for A. We shall refer to the seminal paper [7] as a standard reference for maximal regularity. We also refer the reader to [5, 6, 3] for the L p -maximal regularity of second-order Cauchy problems, to [1, 2] for maximal regularity for non-autonomous problems, to [14, 17] for integro-differential equations and to [4, 13] for fractional differential equations. In this article, we revisit the standard Hilbert space case A = C C under a system perspective: By setting v := Cu we deduce from 1, writing for the time derivative, the operator equation u v 1 u v C u f = v C Now, we ask for the maximal regularity, when the coefficients and are 1 M replaced by more general operators and N acting in space time. Under suitable. 2

3 conditions on M and N, we will show in our main Theorem 2.4, that for a given L 2 -righthand side f, the solution u,v has the following properties. We have that u is weakly C L 2 -differentiable with respect to time and that u,v D. Moreover, the C equation M u u C u f N = v v C v f f is satisfied literally. This remains true if we alter the right-hand side to for any g weakly differentiable g. Our first order approach complements known results on maximal regularity by allowing for quite general coefficients M, N. With this generalization, we enter the realm of so-called evolutionary equations, which we briefly introduce in the next section. This class comprises the standard initial boundary value problems of mathematical physics in a unified setting, we refer to [12] for a survey. After having introduced the mathematical framework, we will provide our main result in Section 2. We conclude this article with several illustrative examples in the last section. The more involved examples are abstract second order problems in both time and space adapted from [3, 14] as well as problems with a fractional time derivative, which is an adaptation from [13]. 1 A brief description of the framework of evolutionary equations We recall the notion of evolutionary equations, as introduced in [8, Solution Theory], a term we use in distinction to classical evolution equations, which are a special case. For this, let throughout be a positive, real parameter and H a Hilbert space. Define L 2 R,H := {f L2 loc R,H t e t ft L 2 R,H}, which endowed with the natural scalar product ˆ f,g := ft,gt H e 2 t dt f,g L 2 R,H is again a Hilbert space. The operator R : D L 2 R,H L2 R,H,f f with f being the distributional derivative and D = {f L 2 R,H f L 2 R,H} defines a normal operator with Re = see e.g [12, Section 2.2]. Indeed, is unitarily equivalent to the operator im of multiplication by the function ξ iξ considered as an operator in L 2 R,H. This spectral representation result is realized by the so-called 3

4 Fourier Laplace transformation L : L 2 R,H L 2 R,H, that is, the unitary extension of the integral operator given by L φξ := 2πˆ 1 e itξ t φt dt ξ R R for bounded, measurable and compactly supported functions φ: R H. In particular, since >, we read off that is boundedly invertible on L2 R,H with 1. It is clear that the spectrum of im is given by the set i[r]. Hence, σ = σim = B C r,r with r = 1/2. Thus, the said spectral representation gives rise to a functional calculus for : Let r > r. For an analytic bounded function M: B C r,r LH we define M := L M im L, where M im φ t := M it φt for allt R andφ L 2 R,H. Again, we refer to [12] for several examples of analytic operator-valued functions of and their occurrence in the context of partial differential equations. The solution theory, that is, unique existence of solutions and continuous dependence on the data, for many linear equations of mathematical physics is covered by the following theorem. For this, note that we do not distinguish between operators defined on H and there respective lifts to the space L 2 R,H. Also the explicit dependence on is frequently suppressed. Theorem 1.1 [8, Solution Theory],[9, Theorem 6.2.5]. Let A: DA H H be skew-selfadjoint, M as above. Assume there is c > such that Re z Mzφ,φ c φ,φ φ H,z B C r,r. Then the operator B := M A defined on its natural domain is closable and the closure is continuously invertible, that is, S := B LL2 R,H. Moreover, S commutes with and for all u DB, and ε >, we have 1ε u DB = D M DA. For the last statement of the theorem one may also consult [16, Lemma 5.2]. We have purposely left out the reference to causality, which also holds and is essential for wellposedness of evolutionary equations in general, but plays a lesser role in this paper. We note the following corollary to Theorem 1.1. Corollary 1.2. With the assumptions and notations in the last theorem, the following is true. Let u DB. If u D M, then u DA and Bu = Bu = M u Au. Proof. Let ε > and define u ε := 1ε u. By Theorem 1.1, we get u ε DB and, since S commutes with, 1ε Bu = Bu ε. Thus, since 1ε 1 as ε 4

5 in the strong operator topology, we infer u ε u and Bu ε Bu in L 2 R,H as ε. Furthermore, from Bu ε = M A u ε = M u ε Au ε = 1ε M uau ε, we read off by the closedness of A, that u DA and Bu = M uau. 2 The main result In this section, we show a maximal regularity result for a prototype equation see also Corollary 2.5 below. Let throughout this section C: DC H H 1 be a densely defined, closed linear operator between Hilbert spaces H and H 1, r >. Moreover, let M: B C r,r LH, N ij : B C r,r LH j,h i analytic and bounded, i,j {,1}. The prototype operator to study in the following is M B := N N 1 N 1 N 11 C C C with domaind M D in the spacel C 2 R,H H 1, where > 1/2r. We will use the following assumptions 1. There is c > such that for all z B C r,r and φ,ψ H H 1 the estimate Mz Re z is satisfied. 2. For some β ], 1] we have N z N 1 z N 1 z N 11 z 2 φ φ φ φ, c ψ ψ, ψ ψ a There is c 1 > such that for all z B C r,r and φ H the estimate Re z β Mzφ,φ c 1 φ,φ is satisfied and the mapping B C r,r z z β Mz is bounded. b If for all z B C r,r, we have N 11 z LH 1 then there is c 2 R such that ψ,ψ Re z β N 11 z c 2 ψ,ψ for all ψ H 1. Some consequences of the latter assumptions are in order. 5

6 Lemma 2.1. Assume that condition 2a holds. Then D M = H β R,H := D β. Proof. We first show M = 1 β M β. Since 1 β M β M and M is closed as a product of a bounded and a closed operator, we get 1 β M β M. Let now u D M and set u ε := 1ε u D D β for ε >. Then u ε u in L 2 R,H as ε and 1 β M β u ε = M u ε = 1ε M u M u ε. Thus, u D 1 β M β with 1 β M β u = M u, which proves the asserted equality. Now, by condition 2a, the operator 1 β M is boundedly invertible on L 2 R,H and hence, the operator 1 β M β is closed. The latter yields 1 β M β = M. But 1 β M is a bounded operator, since z z β Mz is bounded by condition 2a. Hence, D 1 β M β = H β R,H and the assertion follows. Lemma 2.2. Assume condition 1. Then for all z B C r,r, the operator N 11 z is continuously invertible. Proof. The claim is immediate by putting φ,ψ =,ψ in the positivity estimate in condition 1. Lemma 2.3. Let u L 2 R,H. Then u D β if and only if sup ε> β 1ε u <. Proof. From 1ε 1 for all ε >, it follows that u D β is sufficient for the supremum being finite. On the other hand, assume that the supremum is finite. Then there is a sequence ε n n in, such that ε n as n and v := lim n 1ε β n u exists in the weak topology of L 2 R,H. By the weak closedness of β and the fact that 1ε n u u as n, we infer u D. β Theorem 2.4. Assume conditions 1, 2a, and 2b. Then, for each > 1, B given in 2r 2 is continuously invertible on L 2 R,H H 1 and for f,g L 2 R,H H β R,H 1, we have B f,g H β R,H L 2 R,H 1 D C C. 6

7 Proof. We want to apply Corollary 1.2. For this, we have to show that u,v := B M N N f,g D 1 N 1 N 11 By the boundedness of N N 1 N 1 N 11., we are left with showing thatu D M. By Lemma 2.1, we need to show that u D. β Invoking Lemma 2.3, it suffices to show that sup β 1ε u <. ε> So, let ε > and define u ε := 1 ε u. We further set v ε := 1 ε v. By C Theorem 1.1 note that is skew-selfadjoint; and that the needed inequality for C the application of Theorem 1.1 is warranted by 1, we have that M C u ε,v ε DB = D D. C Thus, we read off v ε DC as well as u ε DC D M. Moreover, we have the equalities M u ε N u ε N 1 v ε C v ε = f ε, N 11 v ε N 1 u ε Cu ε = g ε, where f ε := 1ε f and g ε := 1ε g. Next, letting ε in the second equality, we infer by the closedness of C that u DC and Furthermore, we get N 11 v N 1 ucu = g. Cu g N 11 v N 1 u 1 1 N11 c N 1 g f, 3 where we have used condition 1. By Lemma 2.2, we also get v ε = N 11 N 1 u ε Cu ε g ε. Substituting the latter equation into the first one, we arrive at M u ε N u ε N 1 N 11 N 1 u ε Cu ε g ε C N 11 N1 u ε Cu ε g ε = f ε. 7

8 Hence, M u ε = N u ε N 1 N 11 N 1 u ε N 1 N 11 Cuε N 1 N 11 gε C N 11 N1 u ε Cu ε g ε fε. We apply, β u ε L 2 to the latter equation, take real parts and use condition 2a to get c 1 Re β u ε, β u ε Re 1 β M β u ε, β u ε = Re M u ε, u β ε = Re N u ε N 1 N 11 N 1 u ε, β u ε Re N 1 N 11 Cu ε N 1 N 11 g ε, u β ε Re C N 11 N1 u ε Cu ε g ε, β u ε Re f ε, β u ε. We recall that u DC and, hence, u ε DC as well as β u ε DC. Thus, we have c 1 Re u β ε, u β ε Re N u ε N 1 N 11 N1 u ε, u β ε Re N 1 N 11 Cu ε N 1 N 11 g ε, β u ε Re β N11 N1 u ε Cu ε g ε,cuε Re f ε, βu ε. We note that apart from the termre β N11 N1 u ε Cu ε g ε,cuε the remaining terms of the right-hand side can by estimated by K 1 β u ε for some constant K 1, where we also used 3 as well as 1ε 1. For the treatise of Re β N11 N1 u ε Cu ε g ε,cuε we estimate with the 8

9 help of condition 2b note that the implication is not void by Lemma 2.2 Re β N11 N1 u ε Cu ε g ε,cuε = Re β N11 N1 u ε,cu ε Re β N11 Cuε β N11 gε,cu ε N11 = Re uε N 1 β,cu ε Re β N11 N11 Cu ε,cu ε Re gε β,cu ε N 11 N 1 uε β Cu c1 Cu 2 N 11 gε β Cu = N 11 N 1 β Cu c1 u ε Cu 2 N 11 β Cu g K 2 u ε K 3 for somek 2,K 3, where we have again used 3. Hence, we get forp := K 1 K 2 /c and q := K 3 /c that u β 2 ε p β q, u ε which implies Thus, u D β, by Lemma 2.3. β u ε p p q. Another, perhaps more familiar looking, maximal regularity result can now be deduced from Theorem 2.4: Corollary 2.5. Assume conditions 1,2a and 2b to be satisfied, > 1/2r. Then, for all f L 2 R,H, there exists a unique satisfying M un u H β R,H D C N 11 C N 1 u N1 N11 C N 11 C N1 u C N1 u = f. 4 9

10 Proof. Using condition 1, by Theorem 1.1, we infer the existence of a unique v,w L 2 R,H H 1 such that M N N 1 N 1 N 11 C C By Theorem 2.4 and Lemma 2.1, we get v H β R,H w L 2 R,H C 1 D C M = D C D C Hence, f M = M = = N N 1 N 1 N 11 N N 1 N 1 N 11 M v N v N 1 N 1 vn 11 With Lemma 2.2, we obtain from the second line w C w w Cv w = N 11 C N1 v. v f =. w C v C w C v C w. 5 Substituting the latter equation into the first equation of 5, we obtain 4. On the other hand, given u H β R,H D C N 11 C N 1 satisfying 4, we deduce that u N 11 C N1 u is a solution of 5, the solution of which being unique. Thus, the uniqueness statement is also settled. 3 Some Examples Although the strength of the above result lies in the generalty of the material laws accessible, the approach is perhaps best illustrated and by making a link to known results obtained by a different approach. In this spirit, our first example deals with paradigm of maximal regularity, the heat equation, to illustrate the different perspective of our approach on this issue. We then continue with slightly more complex example cases from the literature, which may not be seen to be covered by the general approach developed here. This includes a concluding example for a fractional-in-time evolutionary problem. 1

11 3.1 The heat equation As a warm-up example we consider the paradigmatic case of the heat transport. Let Ω R 3 be a non-empty open where the heat transport is supposed to take place. We consider the equations of heat conduction in the body Ω, which consists of the balance of momentum law ϑdivq = f, where ϑ : R Ω C denotes the temperature density, q : R Ω C 3 stands for the heat flux and f : R Ω C is an external heat source forcing term, and Fourier s law q = kgradϑ, where k LL 2 Ω 3,L 2 Ω 3 is a bounded selfadjoint operator satisfying Re kψ,ψ L 2 Ω 3 c ψ,ψ L 2 Ω 3 ψ L2 Ω 3 for some c >, modeling the heat conductivity of the medium occupying Ω. If we impose suitable boundary conditions, say a homogeneous Dirichlet boundary condition, on ϑ, we end up with the following system 1 k div grad ϑ f =, 6 q where grad is defined as the distributional gradient with domain H 1 Ω and div := grad. Thus, we are indeed in the setting studied in the previous section. Since k is bounded, selfadjoint and strictly positive definite, so is k. Thus, conditions 2a with β = 1 and 1 are clearly satisfied. Moreover, for z B C r,r we have z = it for some > 1,t R and hence, 2r Re z k ψ,ψ = Re it kψ,ψ c ψ,ψ for each ψ L 2 Ω 3, where we have used the selfadjointness of k. This proves that condition 2b is satisfied and thus, Theorem 2.4 yields maximal regularity of 6. In view of Corollary 2.5, we end up with the following result: Corollary 3.1. For all >, f L 2 R,L2 Ω, there exists a unique u H 1 R,L2 Ω 3 Ddivkgrad such that u divkgrad u = f. Remark 3.2. We emphasize that each boundary condition yielding an operator matrix of C the form allows the application of Theorem 2.4. For several examples of such C boundary condition, including mixed and non-local ones we refer to [1]. 11

12 3.2 A second order equation Following [3, Example 6.1], where the much deeper issue of maximal regularity in certain interpolation spaces is addressed, we consider the equation 2 ϑc AB Cϑ = f, where C : DC H H 1 is densely defined closed and linear between the two Hilbert spaces H and H 1, and B LH 1 is selfadjoint, strictly positive definite and A LH 1. First, we note that for > large enough the operator A B = A B = B B A 1 is continuously invertible on L 2 R,H 1, due to a Neumann series argument for this recall that 1/. Hence, setting w := ϑ,q := AB Cϑ, we may rewrite the above problem as a first order equation of the form 1 B A 1 C w f B =. C q Thus, Theorem 2.4 is applicable with the choices Mz = 1, Nz = Az B, g =. Indeed, condition 2a for β = 1 is obviously satisfied while condition 1 follows from Re N 11 zψ,ψ H1 = Re B ψ,ψ Rez B AB AB z 1 ψ,ψ H1 c ψ,ψ B AB 1 z B A ψ,ψ H 1 c B AB 1 r B A ψ,ψ H1 ψ H 1 for z B C r,r with r > small enough which corresponds to > large enough in the argumentation above, where c > is a positive definiteness constant of B, that is, B c. For showing condition 2b for β = 1, we compute Re z Az Bψ,ψ H1 A ψ,ψ 1 2r c ψ,ψ, for each z B C r,r, where we have used the selfadjointness of B and that B c for some c > by assumption. The corresponding statement for the equation, we originally started out with is as follows. Corollary 3.3. There exists > such that for all the following holds: For all f L 2 R,H there exists a unique ϑ H 2 R,H DC AB C satisfying 2 ϑc AB Cϑ = f. Proof. Again, we rely on Corollary 2.5 for β = 1. Note that in the above computations, we used the substitution w = ϑ. We infer that w H 1 R,H, which yields ϑ H 2 R,H. 12

13 3.3 A second order integro-differential equation Let C : DC H H 1 densely defined closed and linear, k : R LH 1 weakly measurable, such that t kt is measurable and k L := kt e t dt < for 1 some >. Moreover, let A,B LH 1 with A selfadjoint and strictly positive definite. We consider the following equation 2 C AB kc u = f, 7 where the convolution operator k is defined by k : L 2 R,H 1 L 2 R,H 1 g t for. By Young s inequality we have that ˆ ksgt s ds k LL 2 R,H 1 k L 1 k L 1 <, so that k is a bounded linear operator on L 2 R,H 1 for each. Moreover, by monotone convergence, we get that limsup k LL 2 R,H 1 lim k L =. For a 1 treatment of integro-differential equations within the framework of evolutionary problems we refer to [15], where this is a special case in the discussion of problems with monotone relations. We rewrite the above problem as a first order problem in the new unknowns v := u and q := A B k Cv. Thus, we arrive at 1 A B k C C v q = f. 8 We note that the operator A B k is indeed boundedly invertible on L 2 R,H 1 for sufficiently large >, since A B k = A 1 A B k and A B k LL 2 R,H 1 1 A B k L < 1 1 for sufficiently large. Moreover, we note that the above problem is an equation of the form discussed in Section 2 with N 11 z := Az B, 2π k iz where k denotes the Fourier-transform of k see [15] for more details. Condition 2a for β = 1 is obviously satisfied in this situation. Moreover, since N 11 z = A A z B k k 2π k iz A k=1 13

14 by Neumann series expansion, we infer that ReN 11 z is uniformly strictly positive definite for z B C 2, for > large enough, since A is strictly positive definite and sup z B C A z B k k 2π k iz A 2, k=1 sup A z B k L A 1. z B C 2, z B k L A 1 This yields that condition 1 is also satisfied. Finally, using the representation z = it for some t R, > large enough, we obtain Rez N 11 z = Rez ARe z B 2π k iz z A B k L 1 c B k L, 1 with c > such that A c. This shows condition 2b β = 1. Thus, Corollary 2.5 applies with β = 1 and yields the maximal regularity of 7. Remark 3.4. The maximal regularity of a similar problem as 7 was studied in [14] in a Banach space setting, where the operators A and B were replaced by real scalars, the kernel k was assumed to be real-valued and the operator C C was replaced by a generator of an analytic semigroup. 3.4 A partial differential equation of fractional type We conclude with the following example taken from [13], where the maximal regularity of the equation β ukau = f has been addressed in spaces of Banach space-valued Hölder continuous functions for some β ],1[. Here A is a sectorial operator and k is a suitable integrable, scalar-valued function, which is supported in the positive reals only. As the case of convolutions has been addressed in the previous two subsections, already, we focus on the simplified equation β uc Cu = f, 9 where C: DC H H 1 is densely defined and closed in the Hilbert spaces H and H 1. We show that the equation 9 admits maximal regularity in L 2 R,H for all >. So, let >. Setting q := Cu, a corresponding 2-by-2 block operator matrix formulation reads β 1 C C u q = f. 14

15 We want to apply Theorem 2.4 or Corollary 2.5 to Mz = z 1 β, Nz =. 1 For this, note that condition 2a is satisfied, since for all r > 1/2, we have Re z β Mzφ,φ = Re z β z 1 β φ,φ = φ,φ z B C r,r,φ H Next, condition 1 follows from [11, Lemma 2.1], which says Re β β. For a proof of condition 2b, we observe that by [11, Lemma 2.1], we have Re Re z β β. Hence, we arrive at the following maximal regularity result for 9. z β = Corollary 3.5. For all >, f L 2 R,H, the equation 9 admits a unique solution u H β R,H DC C. Acknowledgments M. W. carried out this work with financial support of the EPSRC grant EP/L1882/2: Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory. This is gratefully acknowledged. References [1] W. Arendt, R. Chill, S. Fornaro, and C. Poupaud. L p -maximal regularity for nonautonomous evolution equations. J. Differ. Equations, 2371:1 26, 27. [2] W. Arendt, D. Dier, H. Laasri, and E. M. Ouhabaz. Maximal regularity for evolution equations governed by non-autonomous forms. Adv. Differ. Equ., : , 214. [3] C. J. K. Batty, R. Chill, and S. Srivastava. Maximal regularity in interpolation spaces for second-order cauchy problems. In W. Arendt, R. Chill, and Y. Tomilov, editors, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, number 25 in Operator Theory: Advances and Applications, pages Springer International Publishing, 215. [4] S. Bu. Well-posedness of fractional differential equations on vector-valued function spaces. Integral Equations Oper. Theory, 712: , 211. [5] R. Chill and S. Srivastava. L p -maximal regularity for second order Cauchy problems. Math. Z., 2514: ,

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