Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

Bisognin, Vanilde; Buriol, Celene; Ferreira, Marcio V.; Sepúlveda, Mauricio; Vera, Octavio

doi:10.1007/s10440-012-9785-0

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…The Schrödinger equation in moving domains has been widely studied in literature and an example is the classical article of Doescher and Rice [18]. For other references on the subject, we mention [3,5,8,9,17,38,40,43,47,50]. In most of these references, (1.1) is studied in dimension d = 1 or in higher dimensions with symmetries as the radial case or the translating case.…”

Section: Resultsmentioning

confidence: 99%

Schrödinger Equation in Moving Domains

Duca

Joly

2021

Ann. Henri Poincaré

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We consider the Schrödinger equationon Ω(t) ( * )where Ω(t) ⊂ R N is a moving domain depending on the time t ∈ [0, T ]. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain Ω0 and a specific family of unitary maps h (t) : L 2 (Ω(t), C) −→ L 2 (Ω0, C). We show that the conjugation by h provides a new equation of the formwhere h = (h ) −1 . The Hamiltonian H(t) is a magnetic Laplacian operator of the formwhere A is an explicit magnetic potential depending on the deformation of the domain Ω(t). The formulation ( * * ) enables to ensure the existence of weak and strong solutions of the initial problem ( * ) on Ω(t) endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for ( * ) are not the homogeneous but the magnetic ones ∂ν u(t) + i ν|A u(t) = 0, even though ( * ) has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and a time-dependent version of the socalled "Moser's trick". We use this outcome in order to simplify Equation ( * * ) and to guarantee the well-posedness for slightly less regular deformations of Ω(t).

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Section: Resultsmentioning

confidence: 99%

Schrödinger Equation in Moving Domains

Duca

Joly

2021

Ann. Henri Poincaré

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show abstract

“…The Schrödinger equation in moving domains has been widely studied in literature and an example is the classical article of Doescher and Rice [17]. For other references on the subject, we mention [3,8,9,16,29,31,37], while we cite [5,34,40] for the corresponding control problems. In most of these references, (1.1) is studied in dimension d = 1 or in higher dimensions with symmetries as the radial case or the translating case.…”

Section: Resultsmentioning

confidence: 99%

Schrödinger equation in moving domains

Duca,

Joly

2020

Preprint

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We consider the Schrödinger equation i∂tu(t) = −∆u (t) on Ω(t) ( * )where Ω(t) ⊂ R N is a moving domain depending on the time t ∈ [0, T ]. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain Ω0 and a specific family of unitary maps h (t) : L 2 (Ω(t), C) −→ L 2 (Ω0, C). We show that the conjugation by h provides a new equation of the formwhere h = (h ) −1 . The Hamiltonian H(t) is a magnetic Laplacian operator of the formwhere A is an explicit magnetic potential depending on the deformation of the domain Ω(t). The formulation ( * * ) enables to ensure the existence of weak and strong solutions of the initial problem ( * ) on Ω(t) endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for ( * ) are not the homogeneous but the magnetic ones ∂ν u(t) + i ν|A u(t) = 0, even though ( * ) has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and a time-dependent version of the socalled "Moser's trick". We use this outcome in order to simplify Equation ( * * ) and to guarantee the well-posedness for slightly less regular deformations of Ω(t).

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Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges

Knobloch

Krechetnikov

2014

Acta Appl Math

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Asymptotic Behaviour for a Nonlinear Schrödinger Equation in Domains with Moving Boundaries

Cited by 4 publications

References 7 publications

Schrödinger Equation in Moving Domains

Schrödinger Equation in Moving Domains

Schrödinger equation in moving domains

Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges

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