On parabolic final value problems and well-posedness
aa r X i v : . [ m a t h . A P ] F e b ON PARABOLIC FINAL VALUE PROBLEMS AND WELL-POSEDNESS
ANN-EVA CHRISTENSEN A1 AND JON JOHNSEN A A BSTRACT . We prove that a large class of parabolic final value problems is well posed. This resultsvia explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability ofsolutions. This data space is the graph normed domain of an unbounded operator, which represents anew compatibility condition pertinent for final value problems. The framework is evolution equationsfor Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smoothopen set is also covered, and for non-zero Dirichlet data a non-trivial extension of the compatibilitycondition is obtained by addition of an improper Bochner integral.
1. I
NTRODUCTION
Well-posedness of final value problems for a large class of parabolic differential equations is de-scribed 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 , u T ) ∈ Y . This should provide a basic clarification ofa type of problems, which hitherto has been insufficiently understood.As a first example, we characterise the functions u ( t , x ) that, in a C ∞ -smooth bounded open set Ω ⊂ R n with boundary ∂Ω , satisfy the following equations that constitute the final value problem forthe heat equation ( ∆ = ∂ x + · · · + ∂ x n denotes the Laplacian): ∂ t u ( t , x ) − ∆ u ( t , x ) = f ( t , x ) for t ∈ ] , T [ , x ∈ Ω , u ( t , x ) = g ( t , x ) for t ∈ ] , T [ , x ∈ ∂Ω , u ( T , x ) = u T ( x ) for x ∈ Ω . (1)Hereby ( f , g , u T ) are the given data of the problem.In case f = g = u ( t , x ) = e ( T − t ) λ v ( 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 u T equals the eigenfunction v . Our construction includes the set B of such basicsolutions u , its linear hull E = span B and a certain completion E .Using the eigenvalues 0 < λ ≤ λ ≤ . . . and the associated L ( Ω ) -orthonormal basis e , e , . . . of eigenfunctions of − ∆ D , the space E (that corresponds to data u T ∈ span ( e j ) ) clearly consists ofsolutions u being finite sums u ( t , x ) = ∑ j e ( T − t ) λ j ( u T | e j ) e j ( x ) . (2) Mathematics Subject Classification.
Key words and phrases.
Parabolic, final value, compatibility condition, well posed, non-selfadjoint.The second author is supported by the Danish Research Council, Natural Sciences grant no. 4181-00042.
Appeared online in C. R. Acad. Sci. Paris, Ser. I. (2018); DOI: 10.1016/j.crma.2018.01.019 . CHRISTENSEN AND JOHNSEN
So at t = u ( , x ) in L ( Ω ) fulfilling k u ( , · ) k = ∑ j e T λ j | ( u T | e j ) | < ∞ . (3)When summation is extended to all j ∈ N , condition (3) becomes very strong, as it is only satisfiedfor special u T : by Weyl’s law λ j = O ( j / n ) , so a single term in (3) yields | ( u T | e j ) | ≤ c exp ( − T j / n ) ;whence the L -coordinates of such u T decay rapidly for j → ∞ . This has been known since the1950’s; cf. the work of John [Joh55] and Miranker [Mir61].More recently e.g. Isakov [Isa98] emphasized the observation, made already in [Mir61], that (2)gives rise to an instability : the sequence of data data u T , k = e k has length 1 for all k , but (2) gives k u k ( , · ) k = k e T λ k e k k = e T λ k ր ∞ for k → ∞ . Thus (1) is not well-posed in L ( Ω ) .In general this instability shows that the L -norm is an insensitive choice. To obtain well-adaptedspaces for (1) with f = g =
0, one could depart from (3). Indeed, along with the solution space E ,a norm on the final data u T ∈ span ( e j ) can be defined by (3); and ||| u T ||| = ( ∑ ∞ j = e T λ j | ( u T | e j ) | ) / can be used as norm on the u T that correspond to solutions u in the above completion E . This wouldgive well-posedness of the homogeneous version of (1) with u ∈ E . (Cf. [CJ17].)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 thenorm on the data ( f , g , u T ) suitably (cf. ||| u T ||| ), for one has to restrict ( f , g , u T ) to a subspace first byimposing certain compatibility conditions . These have long been known for parabolic problems, butthey have a new form for final value problems.2. T HE ABSTRACT FINAL VALUE PROBLEM
Our 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 Hilbertspaces V ֒ → H ֒ → V ∗ with norms k · k , | · | and k · k ∗ .In this set-up, we consider the following general final value problem: given data f ∈ L ( , T ; V ∗ ) , u T ∈ H , determine the vector distributions u ∈ D ′ ( , T ; V ) fulfilling ∂ t u + Au = f in D ′ ( , T ; V ∗ ) , u ( T ) = u T in H . ) (4)A wealth of parabolic Cauchy problems with homogeneous boundary conditions have been efficientlytreated using such triples ( H , V , a ) and the D ′ ( , T ; V ∗ ) framework in (4); cf. works of Lions andMagenes [LM72], Tanabe [Tan79], Temam [Tem84], Amann [Ama95]. Also recently e.g. Almog,Grebenkov, Helffer, Henry studied variants of the complex Airy operator via such triples [AH15,GHH17, GH16], and our results should at least extend to final value problems for those of theirrealisations that have non-empty spectrum.For the corresponding Cauchy problem we recall that when solving u ′ + Au = f so that u ( ) = u in H , for f ∈ L ( , T ; V ∗ ) , there is a unique solution u in the Banach space X : = L ( , T ; V ) \ C ([ , T ] ; H ) \ H ( , T ; V ∗ ) k u k X = (cid:0) Z T ( k u ( t ) k + k u ′ ( t ) k ∗ ) dt + sup ≤ t ≤ T | u ( t ) | (cid:1) / . (5)For (4) it would therefore be natural to expect solutions u in the same space X . This is correct, butonly when the data ( f , u T ) satisfy substantial further conditions. N FINAL VALUE PROBLEMS AND WELL-POSEDNESS 3
To state these, we utilise that − A generates an analytic semigroup e − tA in B ( H ) and B ( V ∗ ) , andthat e − tA consequently is invertible in the class of closed operators on H , resp. V ∗ ; cf. Proposition 2.2in [CJ17]. Consistently with the case when A generates a group, we set ( e − tA ) − = e tA . (6)Its domain D ( e tA ) = R ( e − tA ) is the Hilbert space normed by k u k = ( | u | + | e tA u | ) / . In the commoncase A has non-empty spectrum, σ ( A ) = /0, there is a chain of strict inclusions D ( e t ′ A ) ( D ( e tA ) ( 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, namely y f = Z T e − ( T − s ) A f ( s ) ds . (8)The map f y f takes values in H , and it is a continuous surjection y f : L ( , 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 , u T ) belong to the subspace Y of L ( , T ; V ∗ ) ⊕ H defined by the conditionu T − y f ∈ D ( e TA ) . (9) In the affirmative case, the solution u is unique in X , and it depends continuously on the data ( f , u T ) in Y , that is k u k X ≤ c k ( f , u T ) k Y , when Y is given the graph norm k ( f , u T ) k Y = (cid:16) | u T | + Z T k f ( t ) k ∗ dt + (cid:12)(cid:12) e TA ( u T − y f ) (cid:12)(cid:12) (cid:17) / . (10)Condition (9) is seemingly a fundamental novelty for the final value problem (4). As for (10), it isthe graph norm of ( f , u T ) e TA ( u T − y f ) , which for Φ ( f , u T ) = u T − y f is the unbounded operator e TA ◦ Φ from L ( , T ; V ∗ ) ⊕ H to H .In fact, e TA Φ is central to a rigorous treatment of (4), for (9) means that e TA Φ must be defined at ( f , u T ) ; i.e. the data space Y is its domain. So since e TA Φ is a closed operator, Y is a Hilbert space,which by (10) is embedded into L ( , T ; V ∗ ) ⊕ H .As an inconvenient aspect, the presence of e − ( T − t ) A and the integration over [ , T ] make (9) non-local in space and time — exacerbated by use of the abstract domain D ( e TA ) , which for larger T givesincreasingly stricter conditions; cf. (7).We regard (9) as a compatibility condition on the data ( f , u T ) , and thus we generalise the notion.Grubb and Solonnikov [GS90] made a systematic treatment of initial -boundary problems of parabolicequations with compatibility conditions, which are necessary and sufficient for well-posedness infull scales of anisotropic L -Sobolev spaces — whereby compatibility conditions are decisive for thesolution’s regularity. In comparison (9) is crucial for the existence question; cf. Theorem 1. Remark . Previously uniqueness was observed by Amann [Ama95, V.2.5.2] in a t -dependent set-up. However, the injectivity of u ( ) u ( T ) was shown much earlier in a set-up with t -dependentsesquilinear forms by Lions and Malgrange [LM60]. Remark . Showalter [Sho74] attempted to characterise the possible u T in terms of Yosida approxima-tions for f = A having half-angle π /
4. As an ingredient the invertibility of analytic semigroupswas claimed by Showalter for such A , but his proof was flawed as A can have semi-angle π / A is not accretive; cf. our example in Remark 3.15 of [CJ17]. CHRISTENSEN AND JOHNSEN
Theorem 1 is proved by considering the full set of solutions to the differential equation u ′ + Au = f .As indicated in (5), for fixed f ∈ L ( , T ; V ∗ ) the solutions in X are parametrised by the initial state u ( ) ∈ H ; and they are also in this set-up necessarily given by the variation of constants formula forthe analytic semigroup e − tA in V ∗ , u ( t ) = e − tA u ( ) + Z t e − ( t − s ) A f ( s ) ds . (11)For t = T this yields a bijective correspondence u ( ) ←→ u ( T ) between the initial and terminalstates—for due to the invertibility of e − TA , cf. (6), one can isolate u ( ) here. Moreover, (11) alsoyields necessity of (9) at once, as the difference u T − y f in (9) must be equal to e − TA u ( ) , whichclearly belongs to the domain D ( e TA ) .Moreover, u ( T ) consists of two radically different parts, cf. (11), even when A is ‘nice’:First, e − tA u ( ) solves the equation for f =
0, and for u ( ) = h ( t ) is strictly convex . Hereby h ( t ) = | e − tA u ( ) | . (12)This results from the injectivity of e − tA when A is normal, or belongs to the class of hyponormaloperators studied by Janas [Jan94], or in case A is accretive — so for such A the complex eigenvalues(if any) cannot give oscillations in the size of e − tA u ( ) , cf. the strict convexity. This stiffness from thestrict convexity is consistent with the fact for analytic semigroups that u ( T ) = e − TA u ( ) is confinedto the dense, but very small space T n ∈ N D ( A n ) .In addition h ( t ) is strictly decreasing with h ′ ( ) ≤ − 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 ( ) = y f : L ( , T ; V ∗ ) → H is surjective, so y f can be anywhere in H . This wasshown with a kind of control-theoretic argument in [CJ17] for the case that A = A ∗ with A − compact;and for general A by using the Closed Range Theorem.Thus the possible final data u T are a sum of an arbitrary y f ∈ H and a term e − TA u ( ) of greatstiffness, so that u T can be prescribed anywhere in the affine space y f + D ( e TA ) . As D ( e TA ) is densein H , and in general there hardly is any control over the direction of y f (if non-zero), it is not feasibleto specify u T a priori in other spaces than H . Instead it is by the condition u T − y f ∈ D ( e TA ) that the u T and f are properly controlled.3. T HE INHOMOGENEOUS HEAT PROBLEM
For general data ( f , g , u T ) in (1), the results in Theorem 1 are applied with A = − ∆ D . The resultsare 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 trickof reducing to boundary data g =
0, for when w ∈ H fulfils w = g = ] , T [ × ∂Ω , then w lacks the regularity needed to test condition (9) on the resulting data ( ˜ f , , ˜ u T ) ofthe reduced problem.Secondly, it therefore takes an effort to show that when the boundary data g =
0, then they do giverise to a correction term z g . This means that condition (9) is replaced by u T − y f + z g ∈ D ( e − T ∆ D ) . (13)Thirdly, because of the low reqularity, it requires some technical diligence to show that, despite thesingularity present in ∆ e ( T − s ) ∆ D at s = T , the correction z g has the structure of an improper Bochner N FINAL VALUE PROBLEMS AND WELL-POSEDNESS 5 integral converging in L ( Ω ) , namely z g = − Z T ∆ e ( T − s ) ∆ D K g ( s ) ds . (14)Hereby the Poisson operator K : H / ( ∂Ω ) → Z ( − ∆ ) is chosen as the inverse of the operator, whichresults by restricting the boundary trace γ : H ( Ω ) → H / ( ∂Ω ) to its coimage Z ( − ∆ ) of harmonicfunctions in H ( Ω ) ; there is a direct sum H ( Ω ) = H ( Ω ) ∔ Z ( − ∆ ) .It is noteworthy that the full influence of the boundary data g on the final state u ( T ) is given in theformula for z g above. In addition z g : H / ( ] , T [ × ∂Ω ) → L ( Ω ) is bounded. Theorem 4.
For given data f ∈ L ( , T ; H − ( Ω )) , g ∈ H / ( ] , T [ × ∂Ω ) , u T ∈ L ( Ω ) the final valueproblem (1) is solved by a function u in the Banach space X , wherebyX = L ( , T ; H ( Ω )) \ C ([ , T ] ; L ( Ω )) \ H ( , T ; H − ( Ω )) , k u k X = (cid:0) Z T ( k u ( t ) k H ( Ω ) + k u ′ ( t ) k H − ( Ω ) ) dt + sup ≤ t ≤ T k u ( t ) k L ( Ω ) (cid:1) / , (15) if and only if the data in terms of (8) and (14) satisfy the compatibility conditionu T − y f + z g ∈ D ( e − T ∆ D ) . (16) In the affirmative case, u is uniquely determined in X and has the representationu ( t ) = e t ∆ D e − T ∆ D ( u T − y f + z g ) + Z t e ( t − s ) ∆ f ( s ) ds − − Z t ∆ e ( t − s ) ∆ D K g ( s ) ds , (17) where the three terms all belong to X as functions of t . Clearly the space of admissible data Y is here a specific subspace of L ( , T ; H − ( Ω )) ⊕ H / ( ] , T [ × ∂Ω ) ⊕ L ( Ω ) , (18)for by setting Φ ( f , g , u T ) = u T − y f + z g we have Y = (cid:8) ( f , g , u T ) | u T − y f + z g ∈ D ( e − T ∆ D ) (cid:9) = D ( e − T ∆ D Φ ) . (19)Here e − T ∆ D Φ is an unbounded operator from the space in (18) to H . Therefore Y is a hilbertableBanach space when endowed with the corresponding graph norm k ( f , g , u T ) k Y = k u T k L ( Ω ) + k g k H / (] , T [ × ∂Ω ) + k f k L ( , T ; H − ( Ω )) + Z Ω (cid:12)(cid:12)(cid:12) e − T ∆ D (cid:0) u T − Z T e − ( T − s ) ∆ f ( s ) ds + − Z T ∆ e ( T − s ) ∆ D K g ( s ) ds (cid:1)(cid:12)(cid:12)(cid:12) dx . (20)Using this the solution operator ( f , g , u T ) u is bounded Y → X , that is, k u k X ≤ c k ( f , g , u T ) k Y . (21)This can be shown by exploiting the bijection u ( ) ←→ u ( T ) to invoke the classical estimates of theinitial value problem, which in the present low regularity setting has no compatibility conditions andtherefore 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 X and Y ; cf. (15) and (18) – (20) . The full proofs of the results in this note can be found in our exposition [CJ17].
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NIT OF E PIDEMIOLOGY AND B IOSTATISTICS , A
ALBORG U NIVERSITY H OSPITAL , H O - BROVEJ
ALBORG , D
ENMARK
E-mail address : [email protected] A D EPARTMENT OF M ATHEMATICAL S CIENCES , A
ALBORG U NIVERSITY , S
KJERNVEJ
ALBORG Ø ST , D ENMARK
E-mail address : (corresponding author): (corresponding author)