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July  2020, 40(7): 4231-4258. doi: 10.3934/dcds.2020179

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

Received  May 2019 Revised  January 2020 Published  April 2020

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.

Citation: Adrien Dekkers, Anna Rozanova-Pierrat, Vladimir Khodygo. Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4231-4258. doi: 10.3934/dcds.2020179
References:
[1]

S. I. AanonsenT. BarkveJ. N. Tjøtta and S. Tjøtta, Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768.   Google Scholar

[2]

R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[3]

S. Alinhac, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.  Google Scholar

[4]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp.  Google Scholar

[5]

N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987.  Google Scholar

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979.  Google Scholar

[7]

P. Caine and M. West, A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165.   Google Scholar

[8]

Z. CaoH. YinL. Zhang and L. Zhu, Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.  doi: 10.1016/S0252-9602(16)30061-3.  Google Scholar

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.  Google Scholar

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[11]

A. Dekkers and A. Rozanova-Pierrat, Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.  doi: 10.3934/dcds.2019012.  Google Scholar

[12]

A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850). Google Scholar

[13]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[14]

G. Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.  doi: 10.1007/BF01762539.  Google Scholar

[15]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.  Google Scholar

[16]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[17]

B. Gustafsson and A. Sundström, Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.  doi: 10.1137/0135030.  Google Scholar

[18] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998.   Google Scholar
[19]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346.  Google Scholar

[20]

D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.  doi: 10.1007/s002050050055.  Google Scholar

[21]

K. Ito, Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323.   Google Scholar

[22]

F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.  Google Scholar

[23]

P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067.  Google Scholar

[24]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[25]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773.  Google Scholar

[26]

B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp. doi: 10.1142/S0218202512500352.  Google Scholar

[27]

B. Kaltenbacher and V. Nikolić, The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532.  Google Scholar

[28]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.  Google Scholar

[29]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.  Google Scholar

[30]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

[31]

V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470.   Google Scholar

[32]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[33]

M. J. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.   Google Scholar

[34]

T. LuoC. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.  Google Scholar

[35]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222.   Google Scholar

[36]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.  Google Scholar

[37]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[38]

B. E. McDonaldP. Caine and M. West, A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167.   Google Scholar

[39]

B. E. McDonald and W. A. Kuperman, Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.  doi: 10.1016/0898-1221(85)90179-8.  Google Scholar

[40]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.  Google Scholar

[41]

A. Rozanova-Pierrat, Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.  doi: 10.1142/S0218202508002863.  Google Scholar

[42]

A. Rozanova-Pierrat, On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.  doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[43]

A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276. Google Scholar

[44]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.  Google Scholar

[45]

T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.  doi: 10.1512/iumj.1991.40.40025.  Google Scholar

[46]

T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp.  Google Scholar

[47]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.  Google Scholar

[48]

M. F. Sukhinin, On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.  doi: 10.1007/BF02069780.  Google Scholar

[49]

J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652. Google Scholar

[50]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537.   Google Scholar

[51]

H. Yin and Q. Qiu, The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.  doi: 10.1007/BF02580410.  Google Scholar

show all references

References:
[1]

S. I. AanonsenT. BarkveJ. N. Tjøtta and S. Tjøtta, Distortion and harmonic generation in the nearfield of a finite amplitude sound beam, The Journal of the Acoustical Society of America, 75 (1984), 749-768.   Google Scholar

[2]

R. A. Adams, Sobolev Spaces, Vol. 65, Pure and Applied Mathematics, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[3]

S. Alinhac, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.  doi: 10.1007/BF01231301.  Google Scholar

[4]

S. Alinhac, A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations, Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002), (2002), Exp. No. Ⅰ, 33 pp.  Google Scholar

[5]

N. S. Bakhvalov, Y. M. Zhileǐkin and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams, American Institute of Physics Translation Series, American Institute of Physics, New York, 1987.  Google Scholar

[6]

L. Bers, F. John and M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, 3A, American Mathematical Society, Providence, RI, 1979.  Google Scholar

[7]

P. Caine and M. West, A tutorial on the non-linear progressive wave equation (NPE). Part 2. Derivation of the three-dimensional cartesian version without use of perturbation expansions, Applied Acoustics, 45 (1995), 155-165.   Google Scholar

[8]

Z. CaoH. YinL. Zhang and L. Zhu, Large time asymptotic behavior of the compressible Navier-Stokes equations in partial space-periodic domains, Acta Math. Sci. Ser. B (Engl. Ed.), 36 (2016), 1167-1191.  doi: 10.1016/S0252-9602(16)30061-3.  Google Scholar

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech., 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.  Google Scholar

[10]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Vol. 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 4$^th$ edition, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-49451-6.  Google Scholar

[11]

A. Dekkers and A. Rozanova-Pierrat, Cauchy problem for the Kuznetsov equation, Discrete Contin. Dyn. Syst., 39 (2019), 277-307.  doi: 10.3934/dcds.2019012.  Google Scholar

[12]

A. Dekkers and A. Rozanova-Pierrat, Models of nonlinear acoustics viewed as an approximation of the Navier-Stokes and Euler compressible isentropicsystems, preprint, arXiv(1811.10850). Google Scholar

[13]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[14]

G. Di Blasio, Linear parabolic evolution equations in $L^p$-spaces, Ann. Mat. Pura Appl., 138 (1984), 55-104.  doi: 10.1007/BF01762539.  Google Scholar

[15]

M. GhisiM. Gobbino and A. Haraux, Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation, Trans. Amer. Math. Soc., 368 (2016), 2039-2079.  doi: 10.1090/tran/6520.  Google Scholar

[16]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[17]

B. Gustafsson and A. Sundström, Incompletely parabolic problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 343-357.  doi: 10.1137/0135030.  Google Scholar

[18] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Academic Press, 1998.   Google Scholar
[19]

D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Rational Mech. Anal., 132 (1995), 1-14.  doi: 10.1007/BF00390346.  Google Scholar

[20]

D. Hoff, Discontinuous solutions of the Navier-Stokes equations for multidimensional flows of heat-conducting fluids, Arch. Rational Mech. Anal., 139 (1997), 303-354.  doi: 10.1007/s002050050055.  Google Scholar

[21]

K. Ito, Smooth global solutions of the two-dimensional Burgers equation, Canad. Appl. Math. Quart., 2 (1994), 283-323.   Google Scholar

[22]

F. John, Nonlinear Wave Equations, Formation of Singularities, Vol. 2, University Lecture Series, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/ulect/002.  Google Scholar

[23]

P. M. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067.  Google Scholar

[24]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.  Google Scholar

[25]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., (2011), 763–773.  Google Scholar

[26]

B. Kaltenbacher, I. Lasiecka and M. K. Pospieszalska, Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 34 pp. doi: 10.1142/S0218202512500352.  Google Scholar

[27]

B. Kaltenbacher and V. Nikolić, The Jordan-Moore-Gibson-Thompson equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time, Math. Models Methods Appl. Sci., 29 (2019), 2523-2556.  doi: 10.1142/S0218202519500532.  Google Scholar

[28]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part Ⅰ. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.  Google Scholar

[29]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; Exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.  Google Scholar

[30]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205.  doi: 10.1007/BF00280740.  Google Scholar

[31]

V. P. Kuznetsov, Equations of nonlinear acoustics, Soviet Phys. Acoust., 16 (1971), 467-470.   Google Scholar

[32]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[33]

M. J. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.   Google Scholar

[34]

T. LuoC. Xie and Z. Xin, Non-uniqueness of admissible weak solutions to compressible Euler systems with source terms, Adv. Math., 291 (2016), 542-583.  doi: 10.1016/j.aim.2015.12.027.  Google Scholar

[35]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part Ⅱ, Acta Acustica United with Acustica, 83 (1997), 197-222.   Google Scholar

[36]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738.  Google Scholar

[37]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.  Google Scholar

[38]

B. E. McDonaldP. Caine and M. West, A tutorial on the nonlinear progressive wave equation (NPE)–part 1, Applied Acoustics, 43 (1994), 159-167.   Google Scholar

[39]

B. E. McDonald and W. A. Kuperman, Time-domain solution of the parabolic equation including nonlinearity, Comput. Math. Appl., 11 (1985), 843-851.  doi: 10.1016/0898-1221(85)90179-8.  Google Scholar

[40]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.  Google Scholar

[41]

A. Rozanova-Pierrat, Qualitative analysis of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, Math. Models Methods Appl. Sci., 18 (2008), 781-812.  doi: 10.1142/S0218202508002863.  Google Scholar

[42]

A. Rozanova-Pierrat, On the derivation of the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and validation of the KZK-approximation for viscous and non-viscous thermo-elastic media, Commun. Math. Sci., 7 (2009), 679-718.  doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[43]

A. Rozanova-Pierrat, Approximation of a compressible Navier-Stokes system by non-linear acoustical models, Proceedings of the International Conference DAYS on DIFFRACTION, (2015), 270–276. Google Scholar

[44]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.  Google Scholar

[45]

T. C. Sideris, The lifespan of smooth solutions to the three-dimensional compressible Euler equations and the incompressible limit, Indiana Univ. Math. J., 40 (1991), 535-550.  doi: 10.1512/iumj.1991.40.40025.  Google Scholar

[46]

T. C. Sideris, The lifespan of 3D compressible flow, Séminaire sur les Équations aux Dérivées Partielles, 5 (1992), 10 pp.  Google Scholar

[47]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.  doi: 10.1353/ajm.1997.0014.  Google Scholar

[48]

M. F. Sukhinin, On the solvability of the nonlinear stationary transport equation, Teoret. Mat. Fiz., 103 (1995), 23-31.  doi: 10.1007/BF02069780.  Google Scholar

[49]

J. N. Tjøtta and S. Tjøtta, Nonlinear equations of acoustics, with application to parametric acoustic arrays, The Journal of the Acoustical Society of America, 69 (1981), 1644–1652. Google Scholar

[50]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537.   Google Scholar

[51]

H. Yin and Q. Qiu, The lifespan for 3-D spherically symmetric compressible sEuler equations, Acta Math. Sinica (N.S.), 14 (1998), 527-534.  doi: 10.1007/BF02580410.  Google Scholar

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} $
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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