December  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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer-Verlag, (2000).  doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[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.  Google Scholar

[10]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[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.  Google Scholar

[12]

R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. Brunnhuber, Well-posedness and Long-Time Behavior of Solutions for the Blackstock-Crighton Equation,, PhD thesis, (2015).   Google Scholar

[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,, , ().   Google Scholar

[17]

J. Burgers, The Nonlinear Diffusion Equation,, Springer, (1974).  doi: 10.1007/978-94-010-1745-9.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., ().   Google Scholar

[26]

G. Crighton David, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11.   Google Scholar

[27]

M. Delfour and J.-P. Zolesio, Shapes and Geometries,, SIAM, (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. Enflo and C. Hedberg, Theory of Nonlinear Acoustics in Fluids,, Fluid Mechanics and Its Applications, (2006).   Google Scholar

[32]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  Google Scholar

[34]

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

[35]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation,, SIAM, (2003).  doi: 10.1137/1.9780898718690.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives,, ESAIM: Control, 14 (2008), 517.  doi: 10.1051/cocv:2008002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy,, submitted., ().   Google Scholar

[52]

B. Kaltenbacher and I. Shevchenko, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions,, submitted., ().   Google Scholar

[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.  Google Scholar

[54]

M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators,, 3rd edition, (2015).   Google Scholar

[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.  Google Scholar

[56]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Research in Applied Mathematics, (1994).   Google Scholar

[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.  Google Scholar

[58]

V. Kuznetsov, Equations of nonlinear acoustics,, Soviet Physics-Acoustics, 16 (1971), 467.   Google Scholar

[59]

I. Lasiecka and W. X., Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

M. Liebler, Modellierung der Dynamischen Wechselwirkung von Hochintensiven Ultraschallfeldern mit Kavitationsblasen,, PhD thesis, (2006).   Google Scholar

[63]

S. Liu and R. Triggiani, An inverse problem for a third order pde arising in high-intensity ultrasound: Global uniqueness and stability by one boundary measurement,, Journal of Inverse and Ill-Posed Problems, 21 (2013), 825.  doi: 10.1515/jip-2012-0096.  Google Scholar

[64]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part I,, Acustica, 82 (1996), 579.   Google Scholar

[65]

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

[66]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part III,, Acustica, 83 (1997), 827.   Google Scholar

[67]

R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability,, Mathematical Methods in the Applied Sciences, 35 (2012), 1896.  doi: 10.1002/mma.1576.  Google Scholar

[68]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Applied Mathematics and Optimization, 64 (2011), 257.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[69]

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

[70]

P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[71]

V. Nikolić, Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions,, J. Math. Anal. Appl., 427 (2015), 1131.  doi: 10.1016/j.jmaa.2015.02.076.  Google Scholar

[72]

V. Nikolić, On Certain Mathematical Aspects of Nonlinear Acoustics: Well-Posedness, Interface Coupling, and Shape Optimization,, PhD thesis, (2015).   Google Scholar

[73]

V. Nikolić and B. Kaltenbacher, On higher regularity for the Westervelt equation with strong nonlinear damping,, submitted and , ().   Google Scholar

[74]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy,, submitted and , ().   Google Scholar

[75]

H. Ockendon and J. Ockendon, Nonlinearity in fluid resonances,, Meccanica, 36 (2001), 297.  doi: 10.1023/A:1013911407811.  Google Scholar

[76]

H. Ockendon and J. Ockendon, Waves and Compressible Flow,, Texts in Applied Mathematics, (2004).   Google Scholar

[77]

A. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications,, Acoustical Society of America, (1989).   Google Scholar

[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,, , ().   Google Scholar

[79]

P. L. Rendón, R. Ezeta and A. Pérez-López, Nonlinear sound propagation in trumpets,, Acta Acustica United with Acustica, 99 (2013), 607.   Google Scholar

[80]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation,, Comptes Rendus Mathematique, 344 (2007), 337.  doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[81]

A. Rozanova-Pierrat, On the controllability for the Khokhlov-Zabolotskaya-Kuznetsov-like equation,, Appl. Anal., 89 (2010), 391.  doi: 10.1080/00036810903569440.  Google Scholar

[82]

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.  doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[83]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The westervelt equation,, Journal of Computational Physics, 302 (2015), 200.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[84]

L. H. Soderholm, Nonlinear acoustics equations to third order - New stabilization of the Burnett equations,, in Mathematical Modeling of Wave Phenomena, (2006), 214.  doi: 10.1063/1.2205805.  Google Scholar

[85]

J. Sokolowski and J. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis,, Springer, (2012).   Google Scholar

[86]

B. Straughan, Acoustic waves in a Cattaneo-Christov gas,, Physics Letters A, 374 (2010), 2667.  doi: 10.1016/j.physleta.2010.04.054.  Google Scholar

[87]

F. Tröltzsch, Optimal Control of Partial Differential Equations,, Theory, (2005).  doi: 10.1090/gsm/112.  Google Scholar

[88]

R. Velasco-Segura and P. Rendón, A finite volume approach for the simulation of nonlinear dissipative acoustic wave propagation,, Wave Motion, 58 (2015), 180.  doi: 10.1016/j.wavemoti.2015.05.006.  Google Scholar

[89]

T. Walsh and M. Torres, Finite element methods for nonlinear acoustics in fluids,, J. Comput. Acoust., 15 (2007), 353.  doi: 10.1142/S0218396X0700338X.  Google Scholar

[90]

D. Wei and P. Jordan, A note on acoustic propagation in power-law fluids: Compact kinks, mild discontinuities, and a connection to finite-scale theory,, International Journal of Non-Linear Mechanics, 48 (2013), 72.  doi: 10.1016/j.ijnonlinmec.2012.07.006.  Google Scholar

[91]

P. J. Westervelt, Parametric acoustic array,, The Journal of the Acoustic Society of America, 35 (1963), 535.  doi: 10.1121/1.1918525.  Google Scholar

[92]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams,, Soviet Physics-Acoustics, 15 (1969), 35.   Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[8]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer-Verlag, (2000).  doi: 10.1007/978-1-4612-1394-9.  Google Scholar

[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.  Google Scholar

[10]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (1991).  doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[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.  Google Scholar

[12]

R. Brunnhuber and P. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. Brunnhuber, Well-posedness and Long-Time Behavior of Solutions for the Blackstock-Crighton Equation,, PhD thesis, (2015).   Google Scholar

[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,, , ().   Google Scholar

[17]

J. Burgers, The Nonlinear Diffusion Equation,, Springer, (1974).  doi: 10.1007/978-94-010-1745-9.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

A. Conejero, C. Lizama and F. Rodenas, Chaotic behaviour of the solutions of the Moore-Gibson- Thompson equation,, Applied Mathematics & Information Sciences., ().   Google Scholar

[26]

G. Crighton David, Model equations of nonlinear acoustics,, Annual Review of Fluid Mechanics, 11 (1979), 11.   Google Scholar

[27]

M. Delfour and J.-P. Zolesio, Shapes and Geometries,, SIAM, (2001).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

B. Enflo and C. Hedberg, Theory of Nonlinear Acoustics in Fluids,, Fluid Mechanics and Its Applications, (2006).   Google Scholar

[32]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  Google Scholar

[34]

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

[35]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation,, SIAM, (2003).  doi: 10.1137/1.9780898718690.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

K. Ito, K. Kunisch and G. H. Peichl, Variational approach to shape derivatives,, ESAIM: Control, 14 (2008), 517.  doi: 10.1051/cocv:2008002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy,, submitted., ().   Google Scholar

[52]

B. Kaltenbacher and I. Shevchenko, Well-posedness of the Westervelt equation with higher order absorbing boundary conditions,, submitted., ().   Google Scholar

[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.  Google Scholar

[54]

M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators,, 3rd edition, (2015).   Google Scholar

[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.  Google Scholar

[56]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, Research in Applied Mathematics, (1994).   Google Scholar

[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.  Google Scholar

[58]

V. Kuznetsov, Equations of nonlinear acoustics,, Soviet Physics-Acoustics, 16 (1971), 467.   Google Scholar

[59]

I. Lasiecka and W. X., Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy,, submitted., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[62]

M. Liebler, Modellierung der Dynamischen Wechselwirkung von Hochintensiven Ultraschallfeldern mit Kavitationsblasen,, PhD thesis, (2006).   Google Scholar

[63]

S. Liu and R. Triggiani, An inverse problem for a third order pde arising in high-intensity ultrasound: Global uniqueness and stability by one boundary measurement,, Journal of Inverse and Ill-Posed Problems, 21 (2013), 825.  doi: 10.1515/jip-2012-0096.  Google Scholar

[64]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part I,, Acustica, 82 (1996), 579.   Google Scholar

[65]

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

[66]

S. Makarov and M. Ochmann, Nonlinear and thermoviscous phenomena in acoustics, part III,, Acustica, 83 (1997), 827.   Google Scholar

[67]

R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order Moore-Gibson-Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability,, Mathematical Methods in the Applied Sciences, 35 (2012), 1896.  doi: 10.1002/mma.1576.  Google Scholar

[68]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation,, Applied Mathematics and Optimization, 64 (2011), 257.  doi: 10.1007/s00245-011-9138-9.  Google Scholar

[69]

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

[70]

P. Neittaanmäki and D. Tiba, Fixed domain approaches in shape optimization problems,, Inverse Problems, 28 (2012).  doi: 10.1088/0266-5611/28/9/093001.  Google Scholar

[71]

V. Nikolić, Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions,, J. Math. Anal. Appl., 427 (2015), 1131.  doi: 10.1016/j.jmaa.2015.02.076.  Google Scholar

[72]

V. Nikolić, On Certain Mathematical Aspects of Nonlinear Acoustics: Well-Posedness, Interface Coupling, and Shape Optimization,, PhD thesis, (2015).   Google Scholar

[73]

V. Nikolić and B. Kaltenbacher, On higher regularity for the Westervelt equation with strong nonlinear damping,, submitted and , ().   Google Scholar

[74]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy,, submitted and , ().   Google Scholar

[75]

H. Ockendon and J. Ockendon, Nonlinearity in fluid resonances,, Meccanica, 36 (2001), 297.  doi: 10.1023/A:1013911407811.  Google Scholar

[76]

H. Ockendon and J. Ockendon, Waves and Compressible Flow,, Texts in Applied Mathematics, (2004).   Google Scholar

[77]

A. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications,, Acoustical Society of America, (1989).   Google Scholar

[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,, , ().   Google Scholar

[79]

P. L. Rendón, R. Ezeta and A. Pérez-López, Nonlinear sound propagation in trumpets,, Acta Acustica United with Acustica, 99 (2013), 607.   Google Scholar

[80]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation,, Comptes Rendus Mathematique, 344 (2007), 337.  doi: 10.1016/j.crma.2007.01.010.  Google Scholar

[81]

A. Rozanova-Pierrat, On the controllability for the Khokhlov-Zabolotskaya-Kuznetsov-like equation,, Appl. Anal., 89 (2010), 391.  doi: 10.1080/00036810903569440.  Google Scholar

[82]

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.  doi: 10.4310/CMS.2009.v7.n3.a9.  Google Scholar

[83]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The westervelt equation,, Journal of Computational Physics, 302 (2015), 200.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[84]

L. H. Soderholm, Nonlinear acoustics equations to third order - New stabilization of the Burnett equations,, in Mathematical Modeling of Wave Phenomena, (2006), 214.  doi: 10.1063/1.2205805.  Google Scholar

[85]

J. Sokolowski and J. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis,, Springer, (2012).   Google Scholar

[86]

B. Straughan, Acoustic waves in a Cattaneo-Christov gas,, Physics Letters A, 374 (2010), 2667.  doi: 10.1016/j.physleta.2010.04.054.  Google Scholar

[87]

F. Tröltzsch, Optimal Control of Partial Differential Equations,, Theory, (2005).  doi: 10.1090/gsm/112.  Google Scholar

[88]

R. Velasco-Segura and P. Rendón, A finite volume approach for the simulation of nonlinear dissipative acoustic wave propagation,, Wave Motion, 58 (2015), 180.  doi: 10.1016/j.wavemoti.2015.05.006.  Google Scholar

[89]

T. Walsh and M. Torres, Finite element methods for nonlinear acoustics in fluids,, J. Comput. Acoust., 15 (2007), 353.  doi: 10.1142/S0218396X0700338X.  Google Scholar

[90]

D. Wei and P. Jordan, A note on acoustic propagation in power-law fluids: Compact kinks, mild discontinuities, and a connection to finite-scale theory,, International Journal of Non-Linear Mechanics, 48 (2013), 72.  doi: 10.1016/j.ijnonlinmec.2012.07.006.  Google Scholar

[91]

P. J. Westervelt, Parametric acoustic array,, The Journal of the Acoustic Society of America, 35 (1963), 535.  doi: 10.1121/1.1918525.  Google Scholar

[92]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams,, Soviet Physics-Acoustics, 15 (1969), 35.   Google Scholar

[1]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[2]

Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361

[3]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[4]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[5]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

[6]

Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377

[7]

Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

[8]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[10]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[11]

Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006

[12]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[13]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[14]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[15]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[16]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[17]

Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations. Electronic Research Archive, 2021, 29 (1) : 1819-1839. doi: 10.3934/era.2020093

[18]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[19]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[20]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (368)
  • HTML views (0)
  • Cited by (19)

Other articles
by authors

[Back to Top]