THE WAVE EQUATION ON A THIN DOMAIN: ENERGY DENSITY AND OBSERVABILITY

We compute the limit energy density of solutions of the linear wave equation in a thin three-dimensional domain, if the wavelength of the Cauchy data is bounded from below by the thickness of the domain. As an application, we obtain a geometric criterion for the uniform observability of solutions of a damped wave equation on such a domain.

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Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891616500235 ◽

2016 ◽

Vol 13 (04) ◽

pp. 833-860

Author(s):

Helge Kristian Jenssen ◽

Charis Tsikkou

Keyword(s):

Cauchy Problem ◽

Wave Equation ◽

Three Dimensional ◽

Initial Boundary ◽

Cauchy Data ◽

Linear Wave ◽

Radial Solutions ◽

Linear Wave Equation ◽

Model Situation ◽

Neumann Solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

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Energy decay rate for a quasi-linear wave equation with localized strong dissipation

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2011.04.064 ◽

2011 ◽

Vol 62 (1) ◽

pp. 164-172 ◽

Cited By ~ 3

Author(s):

Daewook Kim ◽

Yong Han Kang ◽

Mi Jin Lee ◽

Il Hyo Jung

Keyword(s):

Wave Equation ◽

Decay Rate ◽

Energy Decay ◽

Linear Wave ◽

Linear Wave Equation ◽

Energy Decay Rate

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Parameter identification for the linear wave equation with Robin boundary condition

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2017-0093 ◽

2019 ◽

Vol 27 (1) ◽

pp. 25-41

Author(s):

Valeria Bacchelli ◽

Dario Pierotti ◽

Stefano Micheletti ◽

Simona Perotto

Keyword(s):

Wave Equation ◽

Mixed Boundary ◽

Stability Result ◽

Interval Length ◽

Left Endpoint ◽

Linear Wave ◽

Reconstruction Procedure ◽

Element Discretization ◽

Linear Wave Equation ◽

Bounded Interval

Abstract We consider an initial-boundary value problem for the classical linear wave equation, where mixed boundary conditions of Dirichlet and Neumann/Robin type are enforced at the endpoints of a bounded interval. First, by a careful application of the method of characteristics, we derive a closed-form representation of the solution for an impulsive Dirichlet data at the left endpoint, and valid for either a Neumann or a Robin data at the right endpoint. Then we devise a reconstruction procedure for identifying both the interval length and the Robin parameter. We provide a corresponding stability result and verify numerically its performance moving from a finite element discretization.

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Alternative Approach to Infinity

Symposium - International Astronomical Union ◽

10.1017/s0074180900236139 ◽

1974 ◽

Vol 64 ◽

pp. 99-99

Author(s):

Peter G. Bergmann

Keyword(s):

Large Class ◽

Riemannian Manifolds ◽

Quotient Space ◽

Three Dimensional ◽

Cauchy Data ◽

Space Time ◽

Equivalence Classes ◽

Null Infinity ◽

Alternative Approach ◽

Dimensional Domain

Following Penrose's construction of space-time infinity by means of a conformal construction, in which null-infinity is a three-dimensional domain, whereas time- and space-infinities are points, Geroch has recently endowed space-infinity with a somewhat richer structure. An approach that might work with a large class of pseudo-Riemannian manifolds is to induce a topology on the set of all geodesics (whether complete or incomplete) by subjecting their Cauchy data to (small) displacements in space-time and Lorentz rotations, and to group the geodesics all of whose neighborhoods intersect into equivalence classes. The quotient space of geodesics over equivalence classes is to represent infinity. In the case of Minkowski, null-infinity has the usual structure, but I0, I+, and I- each become three-dimensional as well.

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