Models of nonlinear acoustics viewed as an approximation of the Navier–Stokes and Euler compressible isentropic systems

Dekkers, Adrien; Rozanova-Pierrat, Anna

doi:10.4310/cms.2020.v18.n8.a1

Cited by 3 publications

(2 citation statements)

References 31 publications

(97 reference statements)

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“…(iii) While a trace regularity result for the second-order normal derivative cannot be obtained, we wonder whether g ∈ H 2 (Σ) implies ∂w/∂ν ∈ H 1 (Σ). (iv) Instrumental to the study of an inverse problem, the regularity of boundary trace (12) is shown in [32] for the special case of homogeneous boundary data. The said result is contained in our Proposition 6 which pertains to the general non-homogeneous case, and a fortiori in Theorem 1 (a).…”

Section: Remark 2 (Boundary Regularity)mentioning

confidence: 99%

“…(formulated in terms of the acoustic velocity potential ψ); all constants that occurr in the equations are positive: c, b are the speed and diffusivity of sound, ρ > 0 is the mass density, β a > 1. The connections of the Westervelt and Kuznetsov (and Khokhlov-Zabolotskaya-Kuznetsov) models with the Navier-Stokes and Euler compressible system is explored in the recent [12]. A well-recognized issue which arises in the modeling of propagation of acoustic and thermal waves is the paradox of heat conduction, namely, the incongruity between the infinite speed of propagation of a thermal disturbance with the principle of classical mechanics known as causality; see, e.g., [21] and its references.…”

Section: Background Literature Reviewmentioning

confidence: 99%

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The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

Bucci¹,

Eller²

2021

Comptes Rendus. Mathématique

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The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation is analyzed. With the focus on non-homogeneous boundary data, two approaches are offered: one is based on the theory of hyperbolic equations, while the other one uses the theory of operator semigroups. This is a mixed hyperbolic problem with a characteristic spatial boundary. Hence, the regularity results exhibit some deficiencies when compared with the non-characteristic case.Résumé. On analyse le problème de Cauchy-Dirichlet pour l'équation de Moore-Gibson-Thompson avec des données non-homogènes. Deux méthodes sont considérées: la théorie des équations hyperboliques et la théorie des semi-groupes d'opérateurs. Il s'agit d'un problème hyperbolique mixte avec une frontière spatiale caractéristique. Par conséquent, les résultats de régularité présentent certaines lacunes par rapport au cas non caractéristique.

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Section: Remark 2 (Boundary Regularity)mentioning

confidence: 99%

Section: Background Literature Reviewmentioning

confidence: 99%

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

Bucci¹,

Eller²

2021

Comptes Rendus. Mathématique

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Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

2022

Self Cite

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The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in the largest natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in R 2 or R 3 we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.

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Boundary value problems on non-Lipschitz uniform domains: stability, compactness and the existence of optimal shapes

Hinz

Rozanova-Pierrat

Teplyaev

2023

ASY

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We study boundary value problems for bounded uniform domains in R n , n ⩾ 2, with non-Lipschitz, and possibly fractal, boundaries. We prove Poincaré inequalities with uniform constants and trace terms for ( ε , ∞ )-domains contained in a fixed bounded Lipschitz domain. We introduce generalized Dirichlet, Neumann, and Robin problems for Poisson-type equations and prove the Mosco convergence of the associated energy functionals along sequences of suitably converging domains. This implies a stability result for weak solutions, the norm convergence of the associated resolvents, and the convergence of the corresponding eigenvalues and eigenfunctions. We provide compactness results for parametrized classes of admissible domains, energy functionals, and weak solutions. Using these results, we can then prove the existence of optimal shapes in these classes in the sense that they minimize the initially given energy functionals. For the Robin boundary problems, this result is new.

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Models of nonlinear acoustics viewed as an approximation of the Navier–Stokes and Euler compressible isentropic systems

Cited by 3 publications

References 31 publications

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

Boundary value problems on non-Lipschitz uniform domains: stability, compactness and the existence of optimal shapes

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