Characterization of Log-convex decay in non-selfadjoint dynamics

Johnsen, Jon

doi:10.3934/era.2018.25.008

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…1], cf. details on the counter-example in Lemma 3.1 and Remark 3 in [17]). A local version for the Laplacian on R n was given by Rauch [25,Cor.…”

Section: Introductionmentioning

confidence: 99%

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A Class of Well-Posed Parabolic Final Value Problems

Johnsen¹

2020

Applied and Numerical Harmonic Analysis

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This paper focuses on parabolic final value problems, and well-posedness is proved for a large class of these. The clarification is obtained from Hilbert spaces that characterise data that give existence, uniqueness and stability of the solutions. The data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states. It induces a new compatibility condition, depending crucially on the fact that analytic semigroups always are invertible in the class of closed operators. Lax-Milgram operators in vector distribution spaces are the main framework. The final value heat conduction problem on a smooth open set is also proved to be well posed, and non-zero Dirichlet data are shown to require an extended compatibility condition obtained by adding an improper Bochner integral.(1)An area of interest of this could be a nuclear power plant hit by a power failure at time t = 0: after power is regained at t = T > 0, and the reactor temperatures u T (x) are measured, it would be crucial to calculate backwards to settle whether at an earlier time t 0 < T the temperatures u(t 0 , x) could cause damage to the fuel rods. A short plan of the paper is the following: the abstract result on final value problems is given in Section 1.3 below. Its proof then follows in Section 2. The results and proofs for the heat problem in (1) are presented Theorems 5-7 in Section 3.2010 Mathematics Subject Classification. 35A01, 47D06.

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“…1], cf. details on the counter-example in Lemma 3.1 and Remark 3 in [17]). A local version for the Laplacian on R n was given by Rauch [25,Cor.…”

Section: Introductionmentioning

confidence: 99%

“…Because of e −(T −t)A and the integral over [0, T ], condition (20) clearly involves nonlocal operators in both space and time as an inconvenient aspect -which is exacerbated by the abstract domain D(e TA ) that for longer lengths T of the time interval gives increasingly stricter conditions; cf. (17).…”

Section: Introductionmentioning

confidence: 99%

A Class of Well-Posed Parabolic Final Value Problems

Johnsen¹

2020

Applied and Numerical Harmonic Analysis

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“…1]; cf. the details in Lemma 3.1 and Remark 3 in [20]). A local version for the Laplacian on R n was given by Rauch [27,Cor.…”

Section: Preliminaries: Injectivity Of Analytic Semigroupsmentioning

confidence: 99%

“…(15) Whilst this holds if A is self-adjoint or normal, it was emphasized in [5] that it suffices that A is just hyponormal (i.e., D(A) ⊂ D(A * ) and |Ax| ≥ |A * x| for x ∈ D(A), following Janas [18]). Recently this was followed up by the author in [20], where the stronger logarithmic convexity of h(t) was proved equivalent to the formally weaker property of A that, for x ∈ D(A 2 ),…”

Section: Introductionmentioning

confidence: 99%

Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators

Johnsen

2019

era

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This paper treats parabolic final value problems generated by coercive Lax-Milgram operators, and well-posedness is proved for this large class. The result is obtained by means of an isomorphism between Hilbert spaces containing the data and solutions. Like for elliptic generators, the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, and the resulting compatibility condition extends to the coercive context. Lax-Milgram operators in vector distribution spaces is the main framework, but the crucial tool that analytic semigroups always are invertible in the class of closed operators is extended to unbounded semigroups, and this is shown to yield a Duhamel formula for the Cauchy problems in the set-up. The final value heat conduction problem with the homogeneous Neumann boundary condition on a smooth open set is also proved to be well posed in the sense of Hadamard.By definition of Schwartz' vector distribution space D ′ (0, T ;V * ) as the space of continuous linear maps ϕ ∈ C ∞ 0 (]0, T [) → V * , cf.[28], the above equation means that for every scalar test function ϕ ∈ C ∞ 0 (]0, T [) the identity u, −ϕ ′ + Au, ϕ = f , ϕ holds in V * .

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“…Whilst this holds if A is self-adjoint or normal, it was emphasized in [CJ18a] that it suffices that A is just hyponormal (i.e., D(A) ⊂ D(A * ) and |Ax| ≥ |A * x| for x ∈ D(A), following Janas [Jan94]). Recently this was followed up by the author in [Joh18], where the stronger logarithmic convexity of h(t) was proved equivalent to the formally weaker property of A that, for x ∈ D(A 2 ),…”

Section: Theorem 3 ([Joh19b]mentioning

confidence: 99%

Isomorphic well-posedness of the final value problem for the heat equation with the homogeneous Neumann condition

Johnsen¹

2019

Preprint

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This paper concerns the final value problem for the heat equation subjected to the homogeneous Neumann condition on the boundary of a smooth open set in Euclidean space. The problem is here shown to be isomorphically well posed in the sense that there exists a linear homeomorphism between suitably chosen Hilbert spaces containing the solutions and the data, respectively. This improves a recent work of the author, in which the same problem was proven well-posed in the original sense of Hadamard under an additional assumption of Hölder continuity of the source term. Like for its predecessor, the point of departure is an abstract analysis in spaces of vector distributions of final value problems generated by coercive Lax-Milgram operators, yielding isomorphic well-posedness for such problems. Hereby the data space is the graph normed domain of an unbounded operator that maps final states to the corresponding initial states, resulting in a nonlocal compatibility condition on the data. As a novelty, a stronger version of the compatibility condition is introduced with the purpose of characterising the data that yield solutions having the regularity property of being square integrable in the generator's graph norm (instead of in the form domain norm). This result allows a direct application to the class 2 boundary condition in the considered inverse Neumann heat problem.2010 Mathematics Subject Classification. 35A01, 47D06.

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Characterization of Log-convex decay in non-selfadjoint dynamics

Cited by 3 publications

References 14 publications

A Class of Well-Posed Parabolic Final Value Problems

A Class of Well-Posed Parabolic Final Value Problems

Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators

Isomorphic well-posedness of the final value problem for the heat equation with the homogeneous Neumann condition

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