# American Institute of Mathematical Sciences

2015, 4(4): 447-491. doi: 10.3934/eect.2015.4.447

## Mathematics of nonlinear acoustics

 1 Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  May 2015 Revised  October 2015 Published  November 2015

The aim of this paper is to highlight some recent developments and outcomes in the mathematical analysis of partial differential equations describing nonlinear sound propagation. Here the emphasis lies on well-posedness and decay results, first of all for the classical models of nonlinear acoustics, later on also for some higher order models. Besides quoting results, we also try to give an idea on their derivation by showning some of the crucial energy estimates. A section is devoted to optimization problems arising in the practical use of high intensity ultrasound.
While this review puts a certain focus on results obtained in the context of the mentioned FWF project, we also provide some important additional references (although certainly not all of them) for interesting further reading.
Citation: Barbara Kaltenbacher. Mathematics of nonlinear acoustics. Evolution Equations & Control Theory, 2015, 4 (4) : 447-491. doi: 10.3934/eect.2015.4.447
##### References:
 [1] J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints,, Numer. Funct. Anal. Optim., 19 (1998), 697. doi: 10.1080/01630569808816854. [2] H. Amann, Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. [3] H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function spaces,, Jindřich Nečas Center for Mathematical Modeling Lecture Notes, (2009). [4] Y. Angel and C. Aristégui, Weakly nonlinear waves in fluids of low viscosity: Lagrangian and eulerian descriptions,, International Journal of Engineering Science, 74 (2014), 190. doi: 10.1016/j.ijengsci.2013.09.005. [5] A. Bamberger, R. Glowinski and Q. H. Tran, A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change,, SIAM J. Numer. Anal., 34 (1997), 603. doi: 10.1137/S0036142994261518. [6] A. Bermúdez, R. Rodríguez and D. Santamarina, Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations,, J. Comput. Appl. Math., 152 (2003), 17. doi: 10.1016/S0377-0427(02)00694-5. [7] D. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids,, Tech Report GD/E Report GD-1463-52 General Dynamics Corp., (1963), 1463. [8] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer-Verlag, (2000). doi: 10.1007/978-1-4612-1394-9. [9] J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints,, SIAM J. Contr. Opt., 27 (1989), 446. doi: 10.1137/0327023. [10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (1991). doi: 10.1007/978-1-4612-3172-1. [11] R. Brunnhuber, Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation,, J. Math. Anal. Appl., 433 (2016), 1037. doi: 10.1016/j.jmaa.2015.07.046. [12] R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., (). [13] R. Brunnhuber and B. Kaltenbacher, Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt model equation,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 4515. doi: 10.3934/dcds.2014.34.4515. [14] R. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling,, Evolution Equations and Control Theory, 3 (2014), 595. doi: 10.3934/eect.2014.3.595. [15] R. Brunnhuber, Well-posedness and Long-Time Behavior of Solutions for the Blackstock-Crighton Equation,, PhD thesis, (2015). [16] R. Brunnhuber and S. Meyer, Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions,, , (). [17] J. Burgers, The Nonlinear Diffusion Equation,, Springer, (1974). doi: 10.1007/978-94-010-1745-9. [18] E. Casas and F. Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem,, Control and Cybernetics, 31 (2002), 695. [19] M. Chen, M. Torres and T. Walsh, Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation,, Physics Letters A, 373 (2009), 1037. doi: 10.1016/j.physleta.2009.01.042. [20] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems,, Pacific J. Math., 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. [21] C. Christov, On frame indifferent formulation of the maxwell-cattaneo model of finite-speed heat conduction,, Mechanics Research Communications, 36 (2009), 481. doi: 10.1016/j.mechrescom.2008.11.003. [22] I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation,, The Quarterly Journal of Mechanics and Applied Mathematics, 60 (2007), 473. doi: 10.1093/qjmam/hbm017. [23] C. Clason and B. Kaltenbacher, Avoiding degeneracy in the westervelt equation by state constrained optimal control,, Evolution Equations and Control Theory, 2 (2013), 281. doi: 10.3934/eect.2013.2.281. [24] C. Clason, B. Kaltenbacher and S. Veljovic, Boundary optimal control of the Westervelt and the Kuznetsov equation,, J. Math. Anal. Appl., 356 (2009), 738. doi: 10.1016/j.jmaa.2009.03.043. [25] A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., (). [26] G. Crighton David, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. [27] M. Delfour and J.-P. Zolesio, Shapes and Geometries,, SIAM, (2001). [28] M. Delfour, Shape derivatives and differentiability of min max,, in Shape Optimization and Free Boundaries (eds. M. Delfour and G. Sabidussi), (1992), 35. doi: 10.1007/978-94-011-2710-3_2. [29] R. Denk, M. Hieber and J. Prüß, R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type,, Memoirs Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. [30] T. Dreyer, W. Kraus, E. Bauer and R. E. Riedlinger, Investigations of compact focusing transducers using stacked piezoelectric elements for strong sound pulses in therapy,, in Proceedings of the IEEE Ultrasonics Symposium, (2000), 1239. doi: 10.1109/ULTSYM.2000.921547. [31] B. Enflo and C. Hedberg, Theory of Nonlinear Acoustics in Fluids,, Fluid Mechanics and Its Applications, (2006). [32] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [33] B. Flemisch, M. Kaltenbacher and B. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids,, Int. J. Numer. Meth. Engng., 67 (2006), 1791. doi: 10.1002/nme.1669. [34] M. Hamilton and D. Blackstock, Nonlinear Acoustics,, Academic Press, (1998). [35] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation,, SIAM, (2003). doi: 10.1137/1.9780898718690. [36] J. Hoffelner, H. Landes, M. Kaltenbacher and R. Lerch, Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation,, IEEE Transactions on Ultrasonics, 48 (2001), 779. doi: 10.1109/58.920712. [37] K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives for a class of bernoulli problems,, Journal of Mathematical Analysis and Applications, 314 (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [38] K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives,, ESAIM: Control, 14 (2008), 517. doi: 10.1051/cocv:2008002. [39] P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases,, Discrete Contin. Dyn. Syst., 19 (2014), 2189. doi: 10.3934/dcdsb.2014.19.2189. [40] P. Jordan, Nonlinear acoustic phenomena in viscous thermally relaxing fluids: Shock bifurcation and the emergence of diffusive solitons,, J. Acoust. Soc. Am., 124 (2008). doi: 10.1121/1.4782790. [41] P. Jordan, An analytical study of kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation,, Phys. Lett. A, 326 (2004), 77. doi: 10.1016/j.physleta.2004.03.067. [42] P. Jordan, Weakly nonlinear harmonic acoustic waves in classical thermoviscous fluids: A perturbation analysis,, in Proceedings of the OCEANS 2009 MTS/IEEE BILOXI Conference & Exhibition, (2009). [43] B. Kaltenbacher, Boundary observability and stabilization for Westervelt type wave equations,, Applied Mathematics and Optimization, 62 (2010), 381. doi: 10.1007/s00245-010-9108-7. [44] B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation,, Discrete and Continuous Dynamical Systems (DCDS), 2 (2009), 503. doi: 10.3934/dcdss.2009.2.503. [45] B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions,, Discrete and Continuous Dynamical Systems Supplement, 2 (2011), 763. [46] B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay,, Mathematische Nachrichten, 285 (2012), 295. doi: 10.1002/mana.201000007. [47] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equations arising in high intensity ultrasound,, Control and Cybernetics, 40 (2011), 971. [48] B. Kaltenbacher, I. Lasiecka and M. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound,, M3AS, 22 (2012). doi: 10.1142/S0218202512500352. [49] B. Kaltenbacher, I. Lasiecka and S. Veljovic, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data,, in Parabolic problems: Progr. Nonlinear Differential Equations Appl., (2011), 357. doi: 10.1007/978-3-0348-0075-4_19. [50] B. Kaltenbacher, V. Nikolic and M. Thalhammer, Efficient time integration methods based on operator splitting and application to the Westervelt equation,, IMA J. Numer. Anal., 35 (2015), 1092. doi: 10.1093/imanum/dru029. [51] B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy,, submitted., (). [52] B. Kaltenbacher and I. Shevchenko, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions,, submitted., (). [53] B. Kaltenbacher and S. Veljovic, Sensitivity analysis of linear and nonlinear lithotripter models,, European Journal of Applied Mathematics, 22 (2011), 21. doi: 10.1017/S0956792510000276. [54] M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators,, 3rd edition, (2015). [55] R. S. Keiffer, R. McNorton, P. Jordan and I. C. Christov, Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-$\alpha$ model of single-phase lossless fluids,, Wave Motion, 48 (2011), 782. doi: 10.1016/j.wavemoti.2011.04.013. [56] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Research in Applied Mathematics, (1994). [57] N. A. Kudryashov and D. I. Sinelshchikov, Nonlinear waves in bubbly liquids with consideration for viscosity and heat transfer,, Physics Letters A, 374 (2010), 2011. doi: 10.1016/j.physleta.2010.02.067. [58] V. Kuznetsov, Equations of nonlinear acoustics,, Soviet Physics-Acoustics, 16 (1971), 467. [59] I. Lasiecka and W. X., Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy,, submitted., (). [60] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part II: General decay of energy,, Journal of Differential Equations, 259 (2015), 7610. doi: 10.1016/j.jde.2015.08.052. [61] M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall,, Journal of Fluid Mechanics, 31 (1968), 501. doi: 10.1017/S0022112068000303. [62] M. Liebler, Modellierung der Dynamischen Wechselwirkung von Hochintensiven Ultraschallfeldern mit Kavitationsblasen,, PhD thesis, (2006). [63] S. Liu and R. 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Kaltenbacher, On higher regularity for the Westervelt equation with strong nonlinear damping,, submitted and , (). [74] V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy,, submitted and , (). [75] H. Ockendon and J. Ockendon, Nonlinearity in fluid resonances,, Meccanica, 36 (2001), 297. doi: 10.1023/A:1013911407811. [76] H. Ockendon and J. Ockendon, Waves and Compressible Flow,, Texts in Applied Mathematics, (2004). [77] A. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications,, Acoustical Society of America, (1989). [78] A. Rasmussen, M. Sørensen, Y. Gaididei and P. Christiansen, Analytical and numerical modeling of front propagation and interaction of fronts in nonlinear thermoviscous fluids including dissipation,, , (). [79] P. L. Rendón, R. Ezeta and A. Pérez-López, Nonlinear sound propagation in trumpets,, Acta Acustica United with Acustica, 99 (2013), 607. [80] A. 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##### References:
 [1] J.-J. Alibert and J.-P. Raymond, A Lagrange multiplier theorem for control problems with state constraints,, Numer. Funct. Anal. Optim., 19 (1998), 697. doi: 10.1080/01630569808816854. [2] H. Amann, Linear and Quasilinear Parabolic Problems: Volume I: Abstract Linear Theory,, Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. [3] H. Amann, Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function spaces,, Jindřich Nečas Center for Mathematical Modeling Lecture Notes, (2009). [4] Y. Angel and C. Aristégui, Weakly nonlinear waves in fluids of low viscosity: Lagrangian and eulerian descriptions,, International Journal of Engineering Science, 74 (2014), 190. doi: 10.1016/j.ijengsci.2013.09.005. [5] A. Bamberger, R. Glowinski and Q. H. Tran, A domain decomposition method for the acoustic wave equation with discontinuous coefficients and grid change,, SIAM J. Numer. Anal., 34 (1997), 603. doi: 10.1137/S0036142994261518. [6] A. Bermúdez, R. Rodríguez and D. Santamarina, Finite element approximation of a displacement formulation for time-domain elastoacoustic vibrations,, J. Comput. Appl. Math., 152 (2003), 17. doi: 10.1016/S0377-0427(02)00694-5. [7] D. Blackstock, Approximate Equations Governing Finite-Amplitude Sound in Thermoviscous Fluids,, Tech Report GD/E Report GD-1463-52 General Dynamics Corp., (1963), 1463. [8] J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer-Verlag, (2000). doi: 10.1007/978-1-4612-1394-9. [9] J. F. Bonnans and E. Casas, Optimal control of semilinear multistate systems with state constraints,, SIAM J. Contr. Opt., 27 (1989), 446. doi: 10.1137/0327023. [10] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (1991). doi: 10.1007/978-1-4612-3172-1. [11] R. Brunnhuber, Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation,, J. Math. Anal. Appl., 433 (2016), 1037. doi: 10.1016/j.jmaa.2015.07.046. [12] R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., (). [13] R. Brunnhuber and B. Kaltenbacher, Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt model equation,, Discrete and Continuous Dynamical Systems - A, 34 (2014), 4515. doi: 10.3934/dcds.2014.34.4515. [14] R. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling,, Evolution Equations and Control Theory, 3 (2014), 595. doi: 10.3934/eect.2014.3.595. [15] R. Brunnhuber, Well-posedness and Long-Time Behavior of Solutions for the Blackstock-Crighton Equation,, PhD thesis, (2015). [16] R. Brunnhuber and S. Meyer, Optimal regularity and exponential stability for the Blackstock-Crighton equation in $L_p$-spaces with Dirichlet and Neumann boundary conditions,, , (). [17] J. Burgers, The Nonlinear Diffusion Equation,, Springer, (1974). doi: 10.1007/978-94-010-1745-9. [18] E. Casas and F. Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem,, Control and Cybernetics, 31 (2002), 695. [19] M. Chen, M. Torres and T. Walsh, Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation,, Physics Letters A, 373 (2009), 1037. doi: 10.1016/j.physleta.2009.01.042. [20] S. P. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems,, Pacific J. Math., 136 (1989), 15. doi: 10.2140/pjm.1989.136.15. [21] C. Christov, On frame indifferent formulation of the maxwell-cattaneo model of finite-speed heat conduction,, Mechanics Research Communications, 36 (2009), 481. doi: 10.1016/j.mechrescom.2008.11.003. [22] I. Christov, C. I. Christov and P. M. Jordan, Modeling weakly nonlinear acoustic wave propagation,, The Quarterly Journal of Mechanics and Applied Mathematics, 60 (2007), 473. doi: 10.1093/qjmam/hbm017. [23] C. Clason and B. Kaltenbacher, Avoiding degeneracy in the westervelt equation by state constrained optimal control,, Evolution Equations and Control Theory, 2 (2013), 281. doi: 10.3934/eect.2013.2.281. [24] C. Clason, B. Kaltenbacher and S. Veljovic, Boundary optimal control of the Westervelt and the Kuznetsov equation,, J. Math. Anal. Appl., 356 (2009), 738. doi: 10.1016/j.jmaa.2009.03.043. [25] A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., (). [26] G. Crighton David, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11. [27] M. Delfour and J.-P. Zolesio, Shapes and Geometries,, SIAM, (2001). [28] M. Delfour, Shape derivatives and differentiability of min max,, in Shape Optimization and Free Boundaries (eds. M. Delfour and G. Sabidussi), (1992), 35. doi: 10.1007/978-94-011-2710-3_2. [29] R. Denk, M. Hieber and J. Prüß, R-boundedness, Fourier multipliers, and problems of elliptic and parabolic type,, Memoirs Amer. Math. Soc., 166 (2003). doi: 10.1090/memo/0788. [30] T. Dreyer, W. Kraus, E. Bauer and R. E. Riedlinger, Investigations of compact focusing transducers using stacked piezoelectric elements for strong sound pulses in therapy,, in Proceedings of the IEEE Ultrasonics Symposium, (2000), 1239. doi: 10.1109/ULTSYM.2000.921547. [31] B. Enflo and C. Hedberg, Theory of Nonlinear Acoustics in Fluids,, Fluid Mechanics and Its Applications, (2006). [32] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). [33] B. Flemisch, M. Kaltenbacher and B. Wohlmuth, Elasto-acoustic and acoustic-acoustic coupling on nonmatching grids,, Int. J. Numer. Meth. Engng., 67 (2006), 1791. doi: 10.1002/nme.1669. [34] M. Hamilton and D. Blackstock, Nonlinear Acoustics,, Academic Press, (1998). [35] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation,, SIAM, (2003). doi: 10.1137/1.9780898718690. [36] J. Hoffelner, H. Landes, M. Kaltenbacher and R. Lerch, Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation,, IEEE Transactions on Ultrasonics, 48 (2001), 779. doi: 10.1109/58.920712. [37] K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives for a class of bernoulli problems,, Journal of Mathematical Analysis and Applications, 314 (2006), 126. doi: 10.1016/j.jmaa.2005.03.100. [38] K. Ito, K. 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