We prove that a large class of parabolic final value problems is well
posed.This results via explicit Hilbert spaces that characterise the data
yielding existence, uniqueness and stability of solutions. This data space is
the graph normed domain of an unbounded operator, which represents a new
compatibility condition pertinent for final value problems. The framework is
evolution equations for Lax--Milgram operators in vector distribution spaces.
The final value heat equation on a smooth open set is also covered, and for
non-zero Dirichlet data a non-trivial extension of the compatibility condition
is obtained by addition of an improper Bochner integral.Comment: 6 pages. Accepted version; a short announcement of results from our
  full paper on final value problems. Appeared in Comptes Rendu Mathematique

Christensen, Ann-Eva

Johnsen, Jon

English

arXiv

Ann-Eva Christensen

Jon Johnsen

Crossref

Comptes Rendus Mathematique

On parabolic final value problems and well-posedness

We prove that a large class of parabolic final value problems is well posed. This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is that of evolution equations for Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data, a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.</p

Christensen, Ann Eva

VBN (Videnbasen)  Aalborg Universitets forskningsportal

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 356 (2018) 301–305Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial differential equationsOn parabolic final value problems and well-posednessSur les problèmes paraboliques à valeur finale bien posésAnn-Eva Christensen 1, Jon Johnsen 2Department of Mathematics, Aalborg University, Skjernvej 4A, DK-9220 Aalborg Øst, Denmarka r t i c l e i n f o a b s t r a c tArticle history:Received 22 August 2017Accepted after revision 29 January 2018Available online 14 February 2018Presented by the Editorial BoardWe prove that a large class of parabolic final value problems is well posed. This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is that of evolution equations for Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data, a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous prouvons que les problèmes à valeur finale sont bien posés pour une large classe d’opérateurs differentiels paraboliques. Ceci est obtenu via un espace de Hilbert qui caractérise l’existence des données impliquant l’existence, l’unicité et la stabilité des solutions. Cet espace de données est le domaine d’un opérateur non borné muni de la norme du graphe, qui représente une nouvelle condition de compatibilité pertinente pour les problèmes à valeur finale. Le cadre est celui des équations d’évolution pour des opérateurs de Lax–Milgram dans des espaces de distributions vectorielles. Nous étudions aussi le problème à valeur finale pour l’équation de la chaleur sur un ouvert lisse ; pour des données de Dirichlet non nulles, nous obtenons une extension non triviale de la condition de compatibilité par l’addition d’une intégrale de Bochner impropre.© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.E-mail addresses: anneva.ch@gmail.com (A.-E. Christensen), jjohnsen@math.aau.dk (J. Johnsen).1 Present address: Unit of Epidemiology and Biostatistics, Aalborg University Hospital, Hobrovej 18–22, DK-9000 Aalborg, Denmark.2 The second author is supported by the Danish Research Council, Natural Sciences grant No. 4181-00042.https://doi.org/10.1016/j.crma.2018.01.0191631-073X/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.302 A.-E. Christensen, J. Johnsen / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 301–3051. IntroductionWell-posedness of final value problems for a large class of parabolic differential equations is described here. That is, for suitable spaces X , Y specified below, they have existence, uniqueness and stability of solutions u ∈ X for given data ( f , g, uT ) ∈ Y . This should provide a basic clarification of a type of problems, which hitherto has been insufficiently under-stood.As a first example, we characterise the functions u(t, x) that, in a C∞-smooth bounded open set  ⊂ Rn with boundary ∂, satisfy the following equations that constitute the final value problem for the heat equation ( = ∂2x1 +· · ·+ ∂2xn denotes the Laplacian):∂t u(t, x) − u(t, x) = f (t, x) for t ∈ ]0, T [, x ∈ ,u(t, x) = g(t, x) for t ∈ ]0, T [, x ∈ ∂,u(T , x) = uT (x) for x ∈ .⎫⎪⎬⎪⎭ (1)Hereby ( f , g, uT ) are the given data of the problem.In case f = 0, g = 0, the first two lines of (1) are satisfied by u(t, x) = e(T −t)λv(x) for all t ∈ R, if v(x) is an eigenfunction of the Dirichlet realization −D with eigenvalue λ.Thus the homogeneous final value problem (1) has the above u as a basic solution if, coincidentally, the final data uTequals the eigenfunction v . Our construction includes the set B of such basic solutions u, its linear hull E = spanB and a certain completion E .Using the eigenvalues 0 < λ1 ≤ λ2 ≤ . . . and the associated L2()-orthonormal basis e1, e2, . . . of eigenfunctions of −D , the space E (that corresponds to data uT ∈ span(e j)) clearly consists of solutions u being finite sumsu(t, x) = ∑ j e(T −t)λ j (uT | e j)e j(x). (2)So at t = 0 there is, by the finiteness, a vector u(0, x) in L2() fulfilling‖u(0, ·)‖2 = ∑ j e2T λ j |(uT | e j)|2 < ∞. (3)When summation is extended to all j ∈ N, condition (3) becomes very strong, as it is only satisfied for special uT : by Weyl’s law λ j = O( j2/n), so a single term in (3) yields |(uT | e j)| ≤ c exp(−T j2/n); whence the L2-coordinates of such uTdecay rapidly for j → ∞. This has been known since the 1950s; cf. the work of John [9] and Miranker [12].More recently e.g. Isakov [7] emphasized the observation, made already in [12], that (2) gives rise to an instability: the sequence of data uT ,k = ek has length 1 for all k, but (2) gives ‖uk(0, ·)‖ = ‖eT λk ek‖ = eT λk ↗ ∞ for k → ∞. Thus (1) is not well-posed in L2().In general, this instability shows that the L2-norm is an insensitive choice. To obtain well-adapted spaces for (1) with f = 0, g = 0, one could depart from (3). Indeed, along with the solution space E , a norm on the final data uT ∈ span(e j)can be defined by (3); and ‖ |uT ‖ | = (∑∞j=1 e2T λ j |(uT | e j)|2)1/2 can be used as norm on the uT that correspond to solutions u in the above completion E . This would give the well-posedness of the homogeneous version of (1) with u ∈ E . (Cf. [3].)But we have first of all replaced specific eigenvalue distributions by using sesqui-linear forms, cf. Lax–Milgram’s lemma, which allowed us to cover general elliptic operators A.Secondly the fully inhomogeneous problem (1) is covered. Here it does not suffice to choose the norm on the data ( f , g, uT ) suitably (cf. ‖ |uT ‖ |), for one has to restrict ( f , g, uT ) to a subspace first by imposing certain compatibility conditions. These have long been known for parabolic problems, but they have a new form for final value problems.2. The abstract final value problemOur main analysis concerns a (possibly non-selfadjoint) Lax–Milgram operator A defined in H from a bounded V -elliptic sesquilinear form a(·, ·) in a Gelfand triple, i.e. densely injected Hilbert spaces V ↪→ H ↪→ V ∗ with norms ‖ · ‖, | · | and ‖ · ‖∗ .In this set-up, we consider the following general final value problem: given data f ∈ L2(0, T ; V ∗), uT ∈ H , determine the vector distributions u ∈ D ′(0, T ; V ) fulfilling∂t u + Au = f in D ′(0, T ; V ∗),u(T ) = uT in H .}(4)A wealth of parabolic Cauchy problems with homogeneous boundary conditions have been efficiently treated using such triples (H, V , a) and the D ′(0, T ; V ∗) framework in (4); cf. works of Lions and Magenes [10], Tanabe [14], Temam [15], Amann [2]. Also recently, e.g., Almog, Grebenkov, Helffer, Henry studied variants of the complex Airy operator via such triples [1,5,4], and our results should at least extend to final value problems for those of their realisations that have non-empty spectrum.A.-E. Christensen, J. Johnsen / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 301–305 303For the corresponding Cauchy problem, we recall that when solving u′ + Au = f so that u(0) = u0 in H , for f ∈L2(0, T ; V ∗), there is a unique solution u in the Banach spaceX :=L2(0, T ; V )⋂C([0, T ]; H)⋂H1(0, T ; V ∗)‖u‖X =( Tˆ0(‖u(t)‖2 + ‖u′(t)‖2∗)dt + sup0≤t≤T|u(t)|2)1/2. (5)For (4) it would therefore be natural to expect solutions u in the same space X . This is correct, but only when the data ( f , uT ) satisfy substantial further conditions.To state these, we utilise that −A generates an analytic semigroup e−t A in B(H) and B(V ∗), and that e−t A consequently is invertible in the class of closed operators on H , resp. V ∗; cf. Proposition 2.2 in [3]. Consistently with the case when Agenerates a group, we set(e−t A)−1 = et A . (6)Its domain D(et A) = R(e−t A) is the Hilbert space normed by ‖u‖ = (|u|2 +|et A u|2)1/2. In the common case A has non-empty spectrum, σ(A) = ∅, there is a chain of strict inclusionsD(et′ A) D(et A)  H for 0 < t < t′. (7)At the final time, t = T these domains enter the well-posedness result below, where for breviety y f will denote the full yield of the source term f on the system, namelyy f =Tˆ0e−(T −s)A f (s)ds. (8)The map f → y f takes values in H , and it is a continuous surjection y f : L2(0, T ; V ∗) → H .Theorem 1. The final value problem (4) has a solution u in the space X in (5) if and only if the data ( f , uT ) belong to the subspace Yof L2(0, T ; V ∗) ⊕ H defined by the conditionuT − y f ∈ D(eT A). (9)In the affirmative case, the solution u is unique in X, and it depends continuously on the data ( f , uT ) in Y , that is ‖u‖X ≤ c‖( f , uT )‖Y , when Y is given the graph norm‖( f , uT )‖Y =(|uT |2 +Tˆ0‖ f (t)‖2∗ dt +∣∣eT A(uT − y f )∣∣2)1/2. (10)Condition (9) is seemingly a fundamental novelty for the final value problem (4). As for (10), it is the graph norm of ( f , uT ) → eT A(uT − y f ), which for ( f , uT ) = uT − y f is the unbounded operator eT A ◦  from L2(0, T ; V ∗) ⊕ H to H .In fact, eT A is central to a rigorous treatment of (4), for (9) means that eT A must be defined at ( f , uT ); i.e. the data space Y is its domain. So since eT A is a closed operator, Y is a Hilbert space, which by (10) is embedded into L2(0, T ; V ∗) ⊕ H .As an inconvenient aspect, the presence of e−(T −t)A and the integration over [0, T ] make (9) non-local in space and time—exacerbated by use of the abstract domain D(eT A), which for larger T gives increasingly stricter conditions; cf. (7).We regard (9) as a compatibility condition on the data ( f , uT ), and thus we generalise the notion. Grubb and Solon-nikov [6] made a systematic treatment of initial-boundary problems of parabolic equations with compatibility conditions, which are necessary and sufficient for well-posedness in full scales of anisotropic L2-Sobolev spaces—whereby compatibility conditions are decisive for the solution’s regularity. In comparison, (9) is crucial for the existence question; cf. Theorem 1.Remark 2. Previously, uniqueness was observed by Amann [2, V.2.5.2] in a t-dependent set-up. However, the injectivity of u(0) → u(T ) was shown much earlier in a set-up with t-dependent sesquilinear forms by Lions and Malgrange [11].Remark 3. Showalter [13] attempted to characterise the possible uT in terms of Yosida approximations for f = 0 and Ahaving half-angle π/4. As an ingredient, the invertibility of analytic semigroups was claimed by Showalter for such A, but his proof was flawed as A can have semi-angle π/4 even if A2 is not accretive; cf. our example in Remark 3.15 of [3].304 A.-E. Christensen, J. Johnsen / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 301–305Theorem 1 is proved by considering the full set of solutions to the differential equation u′ + Au = f . As indicated in (5), for fixed f ∈ L2(0, T ; V ∗) the solutions in X are parametrised by the initial state u(0) ∈ H ; and they are also in this set-up necessarily given by the variation of constants formula for the analytic semigroup e−t A in V ∗ ,u(t) = e−t Au(0) +tˆ0e−(t−s)A f (s)ds. (11)For t = T , this yields a bijective correspondence u(0) ←→ u(T ) between the initial and terminal states—for due to the in-vertibility of e−T A , cf. (6), one can isolate u(0) here. Moreover, (11) also yields necessity of (9) at once, as the difference uT − y f in (9) must be equal to e−T A u(0), which clearly belongs to the domain D(eT A).Moreover, u(T ) consists of two radically different parts, cf. (11), even when A is ‘nice’:First, e−t Au(0) solves the equation for f = 0, and for u(0) = 0 we obtained in [3] the precise property in non-selfadjoint dynamics that the “height” function h(t) is strictly convex. Herebyh(t) = |e−t Au(0)|. (12)This results from the injectivity of e−t A when A is normal, or belongs to the class of hyponormal operators studied by Janas [8], or in case A2 is accretive — so for such A the complex eigenvalues (if any) cannot give oscillations in the size of e−t Au(0), cf. the strict convexity. This stiffness from the strict convexity is consistent with the fact for analytic semigroups that u(T ) = e−T A u(0) is confined to the dense, but very small space ⋂n∈N D(An).In addition, h(t) is strictly decreasing with h′(0) ≤ −m(A), where m(A) denotes the lower bound; i.e. the short-time behaviour is governed by the numerical range ν(A) also for such A.Secondly, for u(0) = 0 the equation is solved by the integral in (11), which has rather different properties. Its final value y f : L2(0, T ; V ∗) → H is surjective, so y f can be anywhere in H . This was shown with a kind of control-theoretic argument in [3] for the case that A = A∗ with A−1 compact; and for general A by using the Closed Range Theorem.Thus the possible final data uT are a sum of an arbitrary y f ∈ H and a term e−T Au(0) of great stiffness, so that uT can be prescribed anywhere in the affine space y f + D(eT A). As D(eT A) is dense in H , and in general there hardly is any control over the direction of y f (if non-zero), it is not feasible to specify uT a priori in other spaces than H . Instead, it is by the condition uT − y f ∈ D(eT A) that the uT and f are properly controlled.3. The inhomogeneous heat problemFor general data ( f , g, uT ) in (1), the results in Theorem 1 are applied with A = −D . The results are analogous, but less simple to prove and state.First of all, even though it is a linear problem, the compatibility condition (9) destroys the old trick of reducing to boundary data g = 0, for when w ∈ H1 fulfils w = g = 0 on the curved boundary ]0, T [ ×∂, then w lacks the regularity needed to test condition (9) on the resulting data ( f̃ , 0, ̃uT ) of the reduced problem.Secondly, it therefore takes an effort to show that when the boundary data g = 0, then they do give rise to a correction term zg . This means that condition (9) is replaced byuT − y f + zg ∈ D(e−T D ). (13)Thirdly, because of the low regularity, it requires some technical diligence to show that, despite the singularity present in e(T −s)D at s = T , the correction zg has the structure of an improper Bochner integral converging in L2(), namelyzg = −Tˆ0e(T −s)D K0 g(s)ds. (14)Hereby the Poisson operator K0 : H1/2(∂) → Z(−) is chosen as the inverse of the operator, which results by restricting the boundary trace γ0 : H1() → H1/2(∂) to its co-image Z(−) of harmonic functions in H1(); there is a direct sum H1() = H10()  Z(−).It is noteworthy that the full influence of the boundary data g on the final state u(T ) is given in the formula for zgabove. In addition, zg : H1/2( ]0, T [ ×∂) → L2() is bounded.Theorem 4. For given data f ∈ L2(0, T ; H−1()), g ∈ H1/2( ]0, T [ ×∂), uT ∈ L2(), the final value problem (1) is solved by a function u in the Banach space X1, wherebyX1 = L2(0, T ; H1())⋂C([0, T ]; L2())⋂H1(0, T ; H−1()),‖u‖X1 =( Tˆ(‖u(t)‖2H1() + ‖u′(t)‖2H−1())dt + sup0≤t≤T‖u(t)‖2L2())1/2,(15)0A.-E. Christensen, J. Johnsen / C. R. Acad. Sci. Paris, Ser. I 356 (2018) 301–305 305if and only if the data in terms of (8) and (14) satisfy the compatibility conditionuT − y f + zg ∈ D(e−T D ). (16)In the affirmative case, u is uniquely determined in X1 and has the representationu(t) = etD e−T D (uT − y f + zg) +tˆ0e(t−s) f (s)ds − −tˆ0e(t−s)D K0 g(s)ds, (17)where the three terms all belong to X1 as functions of t.Clearly the space of admissible data Y1 is here a specific subspace ofL2(0, T ; H−1()) ⊕ H1/2( ]0, T [×∂) ⊕ L2(), (18)for by setting 1( f , g, uT ) = uT − y f + zg , we haveY1 ={( f , g, uT ) | uT − y f + zg ∈ D(e−T D )}= D(e−T D 1). (19)Here e−T D 1 is an unbounded operator from the space in (18) to H . Therefore Y1 is a hilbertable Banach space when endowed with the corresponding graph norm‖( f , g, uT )‖2Y1 = ‖uT ‖2L2() + ‖g‖2H1/2( ]0,T [ ×∂) + ‖ f ‖2L2(0,T ;H−1())+ˆ∣∣∣e−T D (uT −Tˆ0e−(T −s) f (s)ds + −Tˆ0e(T −s)D K0 g(s)ds)∣∣∣2 dx. (20)Using this the solution operator ( f , g, uT ) → u is bounded Y1 → X1, that is,‖u‖X1 ≤ c‖( f , g, uT )‖Y1 . (21)This can be shown by exploiting the bijection u(0) ←→ u(T ) to invoke the classical estimates of the initial value problem, which in the present low-regularity setting has no compatibility conditions and therefore allows a reduction to the case g = 0. So, in combination with Theorem 4, we have:Theorem 5. The final value Dirichlet heat problem (1) is well posed in the spaces X1 and Y1; cf. (15) and (18)–(20).The full proofs of the results in this note can be found in our exposition [3].References[1] Y. Almog, B. Helffer, On the spectrum of non-selfadjoint Schrödinger operators with compact resolvent, Commun. Partial Differ. Equ. 40 (8) (2015) 1441–1466.[2] H. Amann, Linear and Quasilinear Parabolic Problems, Vol. I, Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, USA, 1995.[3] A.-E. Christensen, J. Johnsen, Final value problems for parabolic differential equations and their well-posedness, preprint, arXiv:1707.02136, 2017.[4] D.S. Grebenkov, B. Helffer, On the spectral properties of the Bloch–Torrey operator in two dimensions, arXiv:1608.01925, 2016.[5] D.S. Grebenkov, B. Helffer, R. Henry, The complex Airy operator on the line with a semipermeable barrier, SIAM J. Math. Anal. 49 (3) (2017) 1844–1894.[6] G. Grubb, V.A. Solonnikov, Solution of parabolic pseudo-differential initial-boundary value problems, J. Differ. Equ. 87 (1990) 256–304.[7] V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, vol. 127, Springer-Verlag, New York, 1998.[8] J. Janas, On unbounded hyponormal operators III, Stud. Math. 112 (1994) 75–82.[9] F. John, Numerical solution of the equation of heat conduction for preceding times, Ann. Mat. Pura Appl. (4) 40 (1955) 129–142.[10] J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, vol. 181, Springer-Verlag, New York–Heidelberg, 1972, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.[11] J.-L. Lions, B. Malgrange, Sur l’unicité rétrograde dans les problèmes mixtes parabolic, Math. Scand. 8 (1960) 227–286.[12] W.L. Miranker, A well posed problem for the backward heat equation, Proc. Amer. Math. Soc. 12 (1961) 243–247.[13] R.E. Showalter, The final value problem for evolution equations, J. Math. Anal. Appl. 47 (1974) 563–572.[14] H. Tanabe, Equations of Evolution, Monographs and Studies in Mathematics, vol. 6, Pitman, Boston, Mass, 1979, translated from the Japanese by N. Mugibayashi and H. Haneda.[15] R. Temam, Navier–Stokes Equations, Theory and Numerical Analysis, 3rd edition, Elsevier Science Publishers B.V., Amsterdam, 1984.

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On parabolic final value problems and well-posedness

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