January 2019, 39(1): 277-307. doi: 10.3934/dcds.2019012

Cauchy problem for the Kuznetsov equation

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

Received  December 2017 Revised  June 2018 Published  October 2018

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a $L^2$-stability estimate, obtained between a regular and a less regular solutions.

Citation: Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012
References:
[1]

R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, 1975.

[2]

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

[3]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, URL http://dx.doi.org/10.1007/978-3-0348-9221-6, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

[4]

H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[5]

R. Chill and S. Srivastava, Lp-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781, URL http://dx.doi.org/10.1007/s00209-005-0815-8.

[6]

M. Ghisi, M. 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, URL http://dx.doi.org/10.1090/tran/6520. doi: 10.1090/tran/6520.

[7]

M. Hamilton and D. Blackstock, Nonlinear Acoustics, Academic Press, 1998.

[8]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1976), 273-294 (1977). doi: 10.1007/BF00251584.

[9]

R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368, URL http://dx.doi.org/10.1016/j.jde.2013.01.023. doi: 10.1016/j.jde.2013.01.023.

[10]

F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series, American Mathematical Society, Providence, RI, 1990, URL http://dx.doi.org/10.1090/ulect/002, Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. doi: 10.1090/ulect/002.

[11]

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

[12]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., 2 (2011), 763-773.

[13]

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, URL http://dx.doi.org/10.1002/mana.201000007. doi: 10.1002/mana.201000007.

[14]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332, URL http://dx.doi.org/10.1002/cpa.3160380305. doi: 10.1002/cpa.3160380305.

[15]

S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space $\mathbb{R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117, URL http://dx.doi.org/10.1002/cpa.3160400105. doi: 10.1002/cpa.3160400105.

[16]

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

[17]

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

[18]

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

[19]

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, URL http://projecteuclid.org/euclid.cms/1256562819. doi: 10.4310/CMS.2009.v7.n3.a9.

[20]

A. Rozanova-Pierrat, Approximation of a Compressible Navier-Stokes System by Non-linear Acoustical Models, Proceedings of the International Conference "Days on Diffraction 2015", St. Petersburg, Russia, IEEE, St. Petersburg, Russia, 2015, URL https://hal.archives-ouvertes.fr/hal-01257919.

[21]

Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226, URL http://dx.doi.org/10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M.

[22]

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

[23]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537, URL http://dx.doi.org/10.1121/1.1918525.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, 1975.

[2]

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

[3]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, URL http://dx.doi.org/10.1007/978-3-0348-9221-6, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

[4]

H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983.

[5]

R. Chill and S. Srivastava, Lp-maximal regularity for second order Cauchy problems, Math. Z., 251 (2005), 751-781, URL http://dx.doi.org/10.1007/s00209-005-0815-8.

[6]

M. Ghisi, M. 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, URL http://dx.doi.org/10.1090/tran/6520. doi: 10.1090/tran/6520.

[7]

M. Hamilton and D. Blackstock, Nonlinear Acoustics, Academic Press, 1998.

[8]

T. J. R. HughesT. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal., 63 (1976), 273-294 (1977). doi: 10.1007/BF00251584.

[9]

R. Ikehata, G. Todorova and B. Yordanov, Wave equations with strong damping in Hilbert spaces, J. Differential Equations, 254 (2013), 3352-3368, URL http://dx.doi.org/10.1016/j.jde.2013.01.023. doi: 10.1016/j.jde.2013.01.023.

[10]

F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series, American Mathematical Society, Providence, RI, 1990, URL http://dx.doi.org/10.1090/ulect/002, Seventh Annual Pitcher Lectures delivered at Lehigh University, Bethlehem, Pennsylvania, April 1989. doi: 10.1090/ulect/002.

[11]

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

[12]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., 2 (2011), 763-773.

[13]

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, URL http://dx.doi.org/10.1002/mana.201000007. doi: 10.1002/mana.201000007.

[14]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332, URL http://dx.doi.org/10.1002/cpa.3160380305. doi: 10.1002/cpa.3160380305.

[15]

S. Klainerman, Remarks on the global Sobolev inequalities in the Minkowski space $\mathbb{R}^{n+1}$, Comm. Pure Appl. Math., 40 (1987), 111-117, URL http://dx.doi.org/10.1002/cpa.3160400105. doi: 10.1002/cpa.3160400105.

[16]

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

[17]

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

[18]

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

[19]

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, URL http://projecteuclid.org/euclid.cms/1256562819. doi: 10.4310/CMS.2009.v7.n3.a9.

[20]

A. Rozanova-Pierrat, Approximation of a Compressible Navier-Stokes System by Non-linear Acoustical Models, Proceedings of the International Conference "Days on Diffraction 2015", St. Petersburg, Russia, IEEE, St. Petersburg, Russia, 2015, URL https://hal.archives-ouvertes.fr/hal-01257919.

[21]

Y. Shibata, On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226, URL http://dx.doi.org/10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M.

[22]

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

[23]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustical Society of America, 35 (1963), 535-537, URL http://dx.doi.org/10.1121/1.1918525.

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