scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

scholarly journals Maximal Regularity for Flexible Structural Systems in Lebesgue Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Claudio Fernández ◽  
Carlos Lizama ◽  
Verónica Poblete
Keyword(s):  
Maximal Regularity ◽  
Flexible Structures ◽  
Fourier Multipliers ◽  
Smooth Domain ◽  
Structural Systems ◽  
Material Damping ◽  
Lebesgue Spaces ◽  
Dirichlet Laplacian ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

We study abstract equations of the formλu′′′(t)+u′′(t)=c2Au(t)+c2μAu′(t)+f(t),0<λ<μwhich is motivated by the study of vibrations of flexible structures possessing internal material damping. We introduce the notion of(α;β;γ)-regularized families, which is a particular case of(a;k)-regularized families, and characterize maximal regularity inLp-spaces based on the technique of Fourier multipliers. Finally, an application with the Dirichlet-Laplacian in a bounded smooth domain is given.

scholarly journals New Explicit Solutions to the Fractional-Order Burgers’ Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. Hafiz Uddin ◽  
Mohammad Asif Arefin ◽  
M. Ali Akbar ◽  
Mustafa Inc
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Acoustic Waves ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Wall Friction ◽  
Weakly Nonlinear ◽  
Bubbly Liquids ◽  
Fractional Differential ◽  
Accuracy Of Results

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables G ′ / G , 1 / G -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of tanh , sech , sinh , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.

Hp-Maximal Regularity and Operator Valued Multipliers on Hardy Spaces

2007 ◽  
Vol 59 (6) ◽  
pp. 1207-1222 ◽  
Author(s):  
Shangquan Bu ◽  
Christian Le Merdy
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Hardy Spaces ◽  
Maximal Regularity ◽  
Type Theorem ◽  
Closed Operator ◽  
Fourier Multipliers ◽  
The Cauchy Problem

AbstractWe consider maximal regularity in the Hp sense for the Cauchy problem u′(t) + Au(t) = f(t) (t ∈ ℝ), where A is a closed operator on a Banach space X and f is an X-valued function defined on ℝ. We prove that if X is an AUMD Banach space, then A satisfies Hp-maximal regularity if and only if A is Rademacher sectorial of type < . Moreover we find an operator A with Hp-maximal regularity that does not have the classical Lp-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces Hp(ℝ X), in the case when X is an AUMD Banach space.

Acoustic Waves of Various Geometry in Multi-Fraction Bubbly Liquids

2018 ◽  
Vol 53 (1) ◽  
pp. 119-126 ◽  
Author(s):  
R. N. Gafiyatov ◽  
D. A. Gubaidullin ◽  
D. D. Gubaidullina
Keyword(s):  
Acoustic Waves ◽  
Bubbly Liquids

The effects of interfacial viscosity on the attenuation and phase speed of acoustic waves in bubbly liquids

2000 ◽  
Vol 107 (5) ◽  
pp. 2866-2866
Author(s):  
Joseph C. Jankovsky ◽  
Ronald A. Roy ◽  
William M. Carey
Keyword(s):  
Acoustic Waves ◽  
Phase Speed ◽  
Bubbly Liquids

scholarly journals Effects of Boundary and Electron Inertia on the Ion Acoustic Wave in a Plasma: A Pseudopotential Approach

1998 ◽  
Vol 51 (1) ◽  
pp. 113 ◽  
Author(s):  
K. K. Mondal ◽  
S. N. Paul ◽  
A. Roy Chowdhury
Keyword(s):  
Acoustic Waves ◽  
Finite Geometry ◽  
Cylindrical Domain ◽  
Plasma Parameters ◽  
Ion Acoustic Waves ◽  
Ion Acoustic Wave ◽  
Electron Inertia ◽  
Ion Acoustic ◽  
Pseudopotential Approach ◽  
Range Of Values

A pseudopotential approach is used to analyse the propagation of ion-acoustic waves in a plasma bounded by a cylindrical domain. The effect of the finite geometry is displayed both analytically and numerically. The phase velocity of the wave is determined and its variation is studied with respect to the plasma parameters. It is observed that the pseudopotential shows a wide variation of shape due to the imposition of a finite boundary condition. It is shown that if the other parameters are kept within a certain range of values, then the trapping of particles is favoured when the presence of the boundary is taken into account.

On the propagation of transient acoustic waves in isothermal bubbly liquids

2006 ◽  
Vol 350 (1-2) ◽  
pp. 56-62 ◽  
Author(s):  
P.M. Jordan ◽  
C. Feuillade
Keyword(s):  
Acoustic Waves ◽  
Bubbly Liquids

Waves of Acoustically Induced Transparency in Bubbly Liquids: Theoretical Prediction and Experimental Validation

2013 ◽  
Author(s):  
Nail A. Gumerov ◽  
Iskander S. Akhatov ◽  
Claus-Dieter Ohl ◽  
Sergei P. Sametov ◽  
Maxim V. Khasimulin ◽  
...  
Keyword(s):  
Acoustic Field ◽  
Acoustic Waves ◽  
Three Dimensional ◽  
Self Organization ◽  
Nonlinear Phenomenon ◽  
Three Dimensional Model ◽  
Bubbly Liquids ◽  
Strongly Nonlinear ◽  
Induced Transparency ◽  
Good Agreement

Self-organization of bubbles in acoustic fields, or self-action of the acoustic waves in bubbly liquids is a strongly nonlinear phenomenon due to two-way interaction of the bubbles and the acoustic field. Theoretical model and preliminary computations predict that waves of self-induced acoustic transparency may exist. Such effect is confirmed in the experiments presented in this paper. Formation of a wave of void fraction which rapidly propagates through the bubbly medium leaving a region almost free of bubbles behind its front is observed in the experiments. Measurements of the dynamics of such a wave at different acoustic frequencies and amplitudes are carried out. A three dimensional model of self-organization of a polydisperse bubble continuum in acoustic field is developed and the results of simulations are compared with experiments. A good agreement of the theory and experiment is found.

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