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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[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.Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

show all references

References:
[1]

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

[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. Google Scholar

[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. Google Scholar

[4]

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

[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.Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

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