Nonlinear Acoustics: Blackstock–Crighton Equations with a Periodic Forcing Term

Celik, Aday; Kyed, Mads

doi:10.1007/s00021-019-0451-4

Cited by 9 publications

(12 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The mathematical analysis of both classical and enhanced models (except for the Burgers equation that can be found in many texbooks on PDEs) is more recent (see, e.g., [23,25,33,34,36] as well as further references in [22]) and to some extent concentrates on initial value problems for these typically second order wave type PDEs. Recently, some interesting results in the time periodic setting have been obtained in, e.g., [9,10].…”

mentioning

confidence: 99%

“…Note that we do not require spatial periodicity here. Also the above mentioned references [9,10] consider time periodic solutions for equations of nonlinear acoustics, namely the Kuznetsov and the Blackstock-Crighton equation, respectively. Their results are based on techniques related to maximal L p regularity for the linearized equation, as opposed to energy estimates in Hilbert Sobolev spaces in our paper.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Periodic solutions and multiharmonic expansions for the Westervelt equation

Kaltenbacher¹

2021

Evolution Equations &Amp; Control Theory

View full text Add to dashboard Cite

In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

show abstract

mentioning

confidence: 99%

mentioning

confidence: 99%

Periodic solutions and multiharmonic expansions for the Westervelt equation

Kaltenbacher¹

2021

Evolution Equations &Amp; Control Theory

View full text Add to dashboard Cite

show abstract

“…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%

Asymptotic behaviors for Blackstock's model of thermoviscous flow

Chen,

Ikehata,

Palmieri

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…Note that this method has also found application in other fields besides fluid dynamics. For example, it was employed by Kyed and Celik [11,12] in order to study damping effects in different nonlinear wave equations, and by Ibrahim, Lemarié-Rieusset and Masmoudi [61] in the investigation of time-periodic solutions to the Navier-Stokes-Maxwell equations.…”

Section: Time-periodic Navier-stokes Equationsmentioning

confidence: 99%

Existence and Spatial Decay of Periodic Navier–Stokes Flows in Exterior Domains

Eiter¹

2020

View full text Add to dashboard Cite

zeigen. Während Lösungen zu diesen Außsenraumproblemen üblicherweise im Rahmen von homogenen Sobolev-Räumen bestimmt werden, ist die Neuheit des vorgestellten Zugangs, dass er einen Rahmen liefert, in dem Lösungen in vollen Sobolev-Räumen nachgewiesen werden.In Kapitel 4 untersuchen wir Problem (NSG) im dreidimensionalen Außenraum Ω unter geeigneten Annahmen an die Daten. Das zugehörige lineare Problem wird in Räumen absolut konvergenter Fourier-Reihen untersucht. Da das zugehörige Resolventenproblem in klassischen Sobolev-Räumen nicht wohlgestellt ist, leiten wir eine lineare Theorie in homogenen Sobolev-Räumen her. Diese neuen Resultate werden angewandt, um Existenz von zeitperiodischen Lösungen des nichtlinearen Systems (NSG) für "kleine" Daten zu zeigen.In Kapitel 5 untersuchen wir das lineare Problem (LNSG) im Ganzraum Ω = R n für allgemeines λ ∈ R, was wir als Problem auf der lokalkompakten abelschen Gruppe T × R n formulieren. Wir führen zeitperiodische Fundamentallösungen für die Lösungen (u, p) dieser Gleichungen ein. Zusätzlich entwickeln wir das Konzept einer zeitperiodischen Fundamentallösung für die Wirbelstärke curl u. Wir untersuchen Integrabilitätseigenschaften und zeigen punktweise Abschätzungen dieser Fundamentallösungen.Das Thema von Kapitel 6 ist die Untersuchung asymptotischer Eigenschaften von zeitperiodischen Lösungen von (NSG) im Falle nichtverschwindender mittlerer Translationsgeschwindigkeit. Das Geschwindigkeitsfeld wird zerlegt in einen zeitunabhängigen Anteil und einen zeitperiodischen Restterm, und wir leiten separate punktweise Abschätzungen für diese beiden Anteile her. Hierdurch entdecken wir neue asymptotische Eigenschaften sowohl des Geschwindigkeitsfelds als auch der Wirbelstärke.

show abstract

Nonlinear Acoustics: Blackstock–Crighton Equations with a Periodic Forcing Term

Cited by 9 publications

References 11 publications

Periodic solutions and multiharmonic expansions for the Westervelt equation

Periodic solutions and multiharmonic expansions for the Westervelt equation

Asymptotic behaviors for Blackstock's model of thermoviscous flow

Existence and Spatial Decay of Periodic Navier–Stokes Flows in Exterior Domains

Contact Info

Product

Resources

About