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Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation

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

    * Corresponding author: Anna Rozanova-Pierrat and Vladimir Khodygo 
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  • 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.

    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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