Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

Adrien Dekkers; Anna Rozanova-Pierrat; Vladimir Khodygo

doi:10.3934/dcds.2020179

Article Contents

2020, Volume 40, Issue 7: 4231-4258. Doi: 10.3934/dcds.2020179

This issue Previous Article Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations Next Article Computability of topological entropy: From general systems to transformations on Cantor sets and the interval

Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

1.
Laboratory Mathématiques et Informatique pour la Complexité et les Systèmes, CentraleSupélec, Univérsité Paris-Saclay, Campus de Gif-sur-Yvette, Plateau de Moulon, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France
2.
Institute of Biological, Environmental and Rural Sciences, Aberystwyth University, Penglais Campus, Aberystwyth, Ceredigion, Wales, SY23 3DA, United Kingdom

^* Corresponding author: Anna Rozanova-Pierrat and Vladimir Khodygo
^* Corresponding author: Anna Rozanova-Pierrat and Vladimir Khodygo

Received: May 2019

Revised: January 2020

Published: April 2020

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Abstract

We relate together different models of non linear acoustic in thermo-elastic media as the Kuznetsov equation, the Westervelt equation, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and the Nonlinear Progressive wave Equation (NPE) and estimate the time during which the solutions of these models keep closed in the $ L^2 $ norm. The KZK and NPE equations are considered as paraxial approximations of the Kuznetsov equation. The Westervelt equation is obtained as a nonlinear approximation of the Kuznetsov equation. Aiming to compare the solutions of the exact and approximated systems in found approximation domains the well-posedness results (for the Kuznetsov equation in a half-space with periodic in time initial and boundary data) are obtained.

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Mathematics Subject Classification: Primary: 35L15, 35L71; Secondary: 35B45.

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Table 1. Approximation results for models derived from the Kuznetsov equation

	KZK		NPE	Westervelt
	periodic boundary condition problem	initial boundary value problem	viscous and inviscid case	viscous case	inviscid case
Theorem	Theorem 2.1	Theorem 2.2	Theorem 3.1	Theorem 4.2
Derivation	paraxial approximation $ u=\Phi(t-\frac{x_1}{c},\varepsilon x_1,\sqrt{\varepsilon} \textbf{x}') $		paraxial approximation $ u=\Psi(\varepsilon t,x_1-ct,\sqrt{\varepsilon} \textbf{x}') $	$ \Pi=u+\frac{1}{c^2}\varepsilon u \partial_t u $
Approximation domain	the half space $ \{x_1>0, x'\in {\mathbb{R}}^{n-1}\} $		$ \mathbb{T}_{x_1}\times\mathbb{R}^2 $	$ \mathbb{R}^n $
Approximation order	$ O(\varepsilon) $		$ O(\varepsilon) $	$ O(\varepsilon^2) $
Estimation	$ \Vert I-I_{aprox}\Vert_{L^2(\mathbb{T}_t\times\mathbb{R}^{n-1})}\leq \varepsilon $ $ z\leq K $	$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon. $ $ t<\frac{T}{\varepsilon} $	$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $ $ t<\frac{T}{\varepsilon} $	$ \Vert (u -\overline{u})_t(t)\Vert_{L^2} $$ + \Vert \nabla (u-\overline{u})(t)\Vert_{L^2} $$ \leq K \varepsilon $ $ t<\frac{T}{\varepsilon} $
Initial data regularity	$ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $ for $ s> \max(\frac{n}{2},2) $	$ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $ for $ \left[\frac{s}{2}\right]>\frac{n}{2}+2 $	$ \xi _0\in H^{s+2}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $ for $ s>\frac{n}{2}+1 $	$ u_0\in H^{s+3}(\mathbb{R}^n) $ $ u_1\in H^{s+3}(\mathbb{R}^3) $ for $ s>\frac{n}{2} $	$ u_0\in H^{s+3}(\mathbb{R}^n) $ $ u_1\in H^{s+2}(\mathbb{R}^3) $ for $ s>\frac{n}{2} $
Data regularity for remainder boundness	$ I_0\in H^{s+\frac{3}{2}}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $ for $ s> \max(\frac{n}{2},2) $	$ I_0\in H^{6}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $ for $ n= 2,3 $, $ I_0\in H^{s}(\mathbb{T}_{t}\times \mathbb{R}^{n-1}_{x'}) $ for $ \left[\frac{s}{2}\right]>\frac{n}{2}+1 $, $ n\geq 4 $	$ \xi _0\in H^{4}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $ for $ n=2,3 $. $ \xi _0\in H^{s}(\mathbb{T}_{x_1}\times \mathbb{R}^{n-1}_{x'}) $ for $ s>\frac{n}{2}+2 $, $ n\geq4 $.	$ u_0\in H^{s+3}(\mathbb{R}^n) $ $ u_1\in H^{s+3}(\mathbb{R}^n) $ for $ s>\frac{n}{2} $	$ u_0\in H^{s+3}(\mathbb{R}^n) $ $ u_1\in H^{s+2}(\mathbb{R}^n) $ for $ s>\frac{n}{2} $

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