The Complex Airy Operator on the Line with a Semipermeable Barrier

Helffer, Bernard; Henry, Raphaël

doi:10.1137/16m1067408

Cited by 26 publications

(83 citation statements)

References 26 publications

(52 reference statements)

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“…In the former papers, the effect of diffusive exchange between compartments onto the MR signal was investigated (see also [18] for a recent overview and references on this topic). In turn, a general asymptotic construction presented in [17] extends and improves the earlier result by de Swiet and Sen for bounded domains. It was shown that, for large enough g, the eigenfunctions of the BT-operator are localized near the boundary points r j at which the normal vector to the boundary is parallel to the gradient direction.…”

supporting

confidence: 78%

“…More recently, the analysis was extended to multiple intervals with semi-permeable boundaries [15,16] and to arbitrary planar domains [17]. In the former papers, the effect of diffusive exchange between compartments onto the MR signal was investigated (see also [18] for a recent overview and references on this topic).…”

mentioning

confidence: 99%

“…In turn, boundary curvature, surface relaxation and membrane permeability were shown to affect the sub-leading terms of order g . For a PGSE experiment with rectangular gradient pulses, the representation (5) can be adapted [15,17].…”

mentioning

confidence: 99%

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Diffusion MRI/NMR at high gradients: Challenges and perspectives

Grebenkov

2018

Microporous and Mesoporous Materials

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We discuss some challenges and recent advances in understanding the macroscopic signal formation at high non-narrow magnetic field gradients at which both the narrow pulse and the Gaussian phase approximations fail. The transverse magnetization and the resulting signal are fully determined by the spectral properties of the non-selfadjoint Bloch-Torrey operator which incorporates the microstructure of a sample through boundary conditions. Since the spectrum of this operator is known to be discrete for isolated pores, the signal can be decomposed onto the eigenmodes of the operator that yields the stretched-exponential decay at high gradients and long times. Moreover, the eigenmodes are localized near specific boundary points that makes the signal more sensitive to the boundaries and thus opens new ways of probing the microstructure. We argue that this behavior is much more general than earlier believed, and should also be valid for unbounded and multi-compartmental domains. In particular, the signal from the extracellular space is not Gaussian at high gradients, in contrast to the common assumption of standard fitting models.

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confidence: 78%

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confidence: 99%

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Diffusion MRI/NMR at high gradients: Challenges and perspectives

Grebenkov

2018

Microporous and Mesoporous Materials

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“…For this the reader may consult the work of Lions and Magenes [8], Tanabe [9], Temam [10], Amann [11]. Also recently, e.g., Almog, Grebenkov, Helffer, Henry studied variants of the complex Airy operator via such triples [12][13][14], and our results should at least extend to final value problems for those of their realisations that have non-empty spectrum. To compare (6) with the analogous Cauchy problem, we recall that whenever u + Au = f is solved under the initial condition u(0) = u 0 ∈ H, for some f ∈ L 2 (0, T; V * ), there is a unique solution u in the Banach space X =L 2 (0, T; V) C([0, T]; H) H 1 (0, T; V * ),…”

Section: The Abstract Final Value Problemmentioning

confidence: 88%

“…Recently Grebenkov, Helffer and Henry [14] studied the complex Airy operator A = −∆ + i x 1 in dimension n = 1. They considered realizations defined on R + by Dirichlet, Neumann and Robin conditions using the Lax-Milgram lemma, so results on boundary homogenous final value problems for − d 2 dx 2 + i x should be straightforward to write down, as in Section 5.1.…”

Section: Remark 18mentioning

confidence: 99%

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Christensen

Johnsen

2018

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This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax-Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

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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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The Complex Airy Operator on the Line with a Semipermeable Barrier

Cited by 26 publications

References 26 publications

Diffusion MRI/NMR at high gradients: Challenges and perspectives

Diffusion MRI/NMR at high gradients: Challenges and perspectives

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

A Class of Well-Posed Parabolic Final Value Problems

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