Geometric perturbation theory and Acoustic Boundary Condition Dynamics

A stability analysis is carried out taking into account slightly porous walls in an idealized detonation confined to a circular pipe. The analysis is carried out using the normal-mode approach and corrections are obtained to the underlying impenetrable wall case results to account for the effect of the slight porosity. The porous walls are modelled by an acoustic boundary condition for the perturbations linking the normal velocity and the pressure components and thus replacing the conventional no-penetration boundary condition at the wall. This new boundary condition necessarily complicates the derivation of the stability problem with respect to the impenetrable wall case. However, exploiting the expressly slight porosity, the modified temporal stability can be determined as a two-point boundary value problem similar to the case of a non-porous wall.

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A note on boundary conditions in Euclidean gravity

Reviews in Mathematical Physics ◽

10.1142/s0129055x21400043 ◽

2021 ◽

pp. 2140004

Author(s):

Edward Witten

Keyword(s):

Boundary Condition ◽

General Relativity ◽

Perturbation Theory ◽

Quantum Gravity ◽

Boundary Conditions ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Conformal Geometry ◽

Extrinsic Curvature ◽

Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

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Robustness of Polynomial Stability of Damped Wave Equations

Journal of Dynamics and Differential Equations ◽

10.1007/s10884-021-10117-y ◽

2022 ◽

Author(s):

Dmytro Baidiuk ◽

Lassi Paunonen

Keyword(s):

Boundary Condition ◽

Wave Equations ◽

Sufficient Conditions ◽

Two Dimensional ◽

Polynomial Stability ◽

One Dimensional ◽

Acoustic Boundary Condition ◽

The Stability ◽

Perturbed Equation ◽

Dimensional Wave

AbstractIn this paper we present new results on the preservation of polynomial stability of damped wave equations under addition of perturbing terms. We in particular introduce sufficient conditions for the stability of perturbed two-dimensional wave equations on rectangular domains, a one-dimensional weakly damped Webster’s equation, and a wave equation with an acoustic boundary condition. In the case of Webster’s equation, we use our results to compute explicit numerical bounds that guarantee the polynomial stability of the perturbed equation.

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