Well-posedness for the abstract Blackstock–Crighton–Westervelt equation

Gambera, Laura R.; Lizama, Carlos; Prokopczyk, Andréa

doi:10.1007/s00028-020-00580-3

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Therefore, by using Lemma 2.1 and combining ( 16), ( 17), (18) and the above estimates for ψ t , ψ tt in the different zones of the phase space, the following propositions for the energy estimates hold.…”

Section: − L 2 Decay Estimates With Additional L 1 Regularitymentioning

confidence: 90%

“…Concerning some studies on linear or nonlinear Blackstock's model under Becker's assumption (cf. next subsection), we refer the interested readers to [9,6,7,10,29,11,18] for Dirichlet or Neumann boundary value problems.…”

Section: Nonlinear Effects In Acousticsmentioning

confidence: 99%

“…To end the proof, we just need to consider some estimates for bounded frequencies. From (18) we get exponential decay estimates without additional assumption on the regularity of initial data. Remark 2.2.…”

Section: − L 2 Decay Estimates With Additional Weighted L 1 Regularitymentioning

confidence: 99%

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Asymptotic behaviors for Blackstock's model of thermoviscous flow

Chen,

Ikehata,

Palmieri

2021

Preprint

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We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space R n . This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive L 2 estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case n 5 and the optimal growth rate for the L 2 -norm of the solution for n = 3, 4; the singular limit problem in determining the first-and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model.

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Section: − L 2 Decay Estimates With Additional L 1 Regularitymentioning

confidence: 90%

Section: Nonlinear Effects In Acousticsmentioning

confidence: 99%

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Asymptotic behaviors for Blackstock's model of thermoviscous flow

Chen,

Ikehata,

Palmieri

2021

Preprint

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“…Concerning previous researches on the Blackstock-Crighton equation under Becker's assumption, we refer the interested readers to [6,3,4,7,8,16] for Dirichlet or Neumann boundary value problems. Additionally, replacing Fourier's law of heat conduction by Cattaneo's law in the energy conservation of modeling, [22] lately investigated global (in time) existence of small data Sobolev solution to Blackstock-Cattaneo model in the whole space, i.e.…”

Section: Background Of Blackstock's Modelmentioning

confidence: 99%

Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow

Chen¹,

Takeda²

2022

Preprint

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In the classical theory of acoustic waves, Blackstock's model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate large-time asymptotic behaviors of the linear Cauchy problem for general Blackstock's model (that is, without Becker's assumption on monatomic perfect gases). We derive first-and second-order asymptotic profiles of solution as t ≫ 1 by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in [Chen-Ikehata-Palmieri, Indiana Univ. Math. J. (2023)] for lower dimensional cases, but also illustrate the optimal leading term and novel second-order profiles of solution with additional weighted L 1 data.

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Large-Time Asymptotic Behaviors for Linear Blackstock’s Model of Thermoviscous Flow

Chen

Takeda

2023

Appl Math Optim

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Well-posedness for the abstract Blackstock–Crighton–Westervelt equation

Cited by 5 publications

References 21 publications

Asymptotic behaviors for Blackstock's model of thermoviscous flow

Asymptotic behaviors for Blackstock's model of thermoviscous flow

Large-time asymptotic behaviors for linear Blackstock's model of thermoviscous flow

Large-Time Asymptotic Behaviors for Linear Blackstock’s Model of Thermoviscous Flow

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