Well‐Posedness, Long‐Time Behavior, and Discretization of Some Models of Nonlinear Acoustics in Velocity–Enthalpy Formulation
Authors
Abstract
We study a class of models for nonlinear acoustics, including the well‐known Westervelt and Kuznetsov equations, as well as a model of Rasmussen that can be seen as a thermodynamically consistent modification of the latter. Using linearization, energy estimates, and fixed‐point arguments, we establish the existence and uniqueness of solutions that, for sufficiently small data, are global in time and converge exponentially fast to equilibrium. In contrast to previous work, our analysis is based on a velocity–enthalpy formulation of the problem, whose weak form reveals the underlying port‐Hamiltonian structure. Moreover, the weak form of the problem is particularly well suited for a structure‐preserving discretization. This is demonstrated in numerical tests, which also highlight typical characteristics of the models under consideration.
Citation
- Journal: Mathematical Methods in the Applied Sciences
- Year: 2025
- Volume:
- Issue:
- Pages:
- Publisher: Wiley
- DOI: 10.1002/mma.10753
BibTeX
@article{Egger_2025,
title={{Well‐Posedness, Long‐Time Behavior, and Discretization of Some Models of Nonlinear Acoustics in Velocity–Enthalpy Formulation}},
ISSN={1099-1476},
DOI={10.1002/mma.10753},
journal={Mathematical Methods in the Applied Sciences},
publisher={Wiley},
author={Egger, Herbert and Fritz, Marvin},
year={2025}
}
References
- Nonlinear Acoustics. (Springer Nature Switzerland, 2024). doi:10.1007/978-3-031-58963-8 – 10.1007/978-3-031-58963-8
- Kaltenbacher, M. Numerical Simulation of Mechatronic Sensors and Actuators. (Springer Berlin Heidelberg, 2015). doi:10.1007/978-3-642-40170-1 – 10.1007/978-3-642-40170-1
- Jordan, P. M. A survey of weakly-nonlinear acoustic models: 1910–2009. Mechanics Research Communications vol. 73 127–139 (2016) – 10.1016/j.mechrescom.2016.02.014
- Christov, I., Christov, C. I. & Jordan, P. M. Modeling weakly nonlinear acoustic wave propagation. The Quarterly Journal of Mechanics and Applied Mathematics vol. 60 473–495 (2007) – 10.1093/qjmam/hbm017
- Abramov, O. V. High-Intensity Ultrasonics. (CRC Press, 2019). doi:10.1201/9780203751954 – 10.1201/9780203751954
- Dreyer, T., Krauss, W., Bauer, E. & Riedlinger, R. E. Investigations of compact self focusing transducers using stacked piezoelectric elements for strong sound pulses in therapy. 2000 IEEE Ultrasonics Symposium. Proceedings. An International Symposium (Cat. No.00CH37121) vol. 2 1239–1242 – 10.1109/ultsym.2000.921547
- Hoffelner, J., Landes, H., Kaltenbacher, M. & Lerch, R. Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control vol. 48 779–786 (2001) – 10.1109/58.920712
- Kaltenbacher, M., Landes, H., Hoffelner, J. & Simkovics, R. Use of modern simulation for industrial applications of high power ultrasonics. 2002 IEEE Ultrasonics Symposium, 2002. Proceedings. 673–678 doi:10.1109/ultsym.2002.1193491 – 10.1109/ultsym.2002.1193491
- Rasmussen, A. R., Sørensen, M. P., Gaididei, Yu. B. & Christiansen, P. L. Analytical and Numerical Modelling of Thermoviscous Shocks and Their Interactions in Nonlinear Fluids Including Dissipation. Mathematics in Industry 997–1002 (2010) doi:10.1007/978-3-642-12110-4_159 – 10.1007/978-3-642-12110-4_159
- Rassmusen, A. R., Sørensen, M. P., Gaididei, Y. B. & Christiansen, P. L. Interacting Wave Fronts and Rarefaction Waves in a Second Order Model of Nonlinear Thermoviscous Fluids. Acta Applicandae Mathematicae vol. 115 43–61 (2010) – 10.1007/s10440-010-9581-7
- Kaltenbacher B., Well‐Posedness of the Westervelt and the Kuznetsov Equation With Nonhomogeneous Neumann Boundary Conditions. Conference Publications (2011)
- Kaltenbacher, B. & Lasiecka, I. An analysis of nonhomogeneous Kuznetsov’s equation: Local and global well‐posedness; exponential decay. Mathematische Nachrichten vol. 285 295–321 (2011) – 10.1002/mana.201000007
- Kaltenbacher, B., Lasiecka, I. & Veljović, S. Well-posedness and Exponential Decay for the Westervelt Equation with Inhomogeneous Dirichlet Boundary Data. Progress in Nonlinear Differential Equations and Their Applications 357–387 (2011) doi:10.1007/978-3-0348-0075-4_19 – 10.1007/978-3-0348-0075-4_19
- Kaltenbacher, B. & Thalhammer, M. Fundamental models in nonlinear acoustics part I. Analytical comparison. Mathematical Models and Methods in Applied Sciences vol. 28 2403–2455 (2018) – 10.1142/s0218202518500525
- Meyer, S. & Wilke, M. Optimal Regularity and Long-Time Behavior of Solutions for the Westervelt Equation. Applied Mathematics & Optimization vol. 64 257–271 (2011) – 10.1007/s00245-011-9138-9
- Meyer, S. & Wilke, M. Global well-posedness and exponential stability for Kuznetsov’s equation in $L_p$-spaces. Evolution Equations & Control Theory vol. 2 365–378 (2013) – 10.3934/eect.2013.2.365
- Tani, A. Mathematical analysis in nonlinear acoustics. AIP Conference Proceedings vol. 1903 020003 (2017) – 10.1063/1.5012614
- Fritz, M., Nikolić, V. & Wohlmuth, B. Well-posedness and numerical treatment of the Blackstock equation in nonlinear acoustics. Mathematical Models and Methods in Applied Sciences vol. 28 2557–2597 (2018) – 10.1142/s0218202518500550
- Kaltenbacher, B., Meliani, M. & Nikolić, V. The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation. Applied Mathematics & Optimization vol. 89 (2024) – 10.1007/s00245-024-10130-9
- Marchand, R., McDevitt, T. & Triggiani, R. 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 vol. 35 1896–1929 (2012) – 10.1002/mma.1576
- Tu, Z. & Liu, W. Well‐posedness and exponential decay for the Moore–Gibson–Thompson equation with time‐dependent memory kernel. Mathematical Methods in the Applied Sciences vol. 46 10465–10479 (2023) – 10.1002/mma.9133
- Nikolić, V. & Wohlmuth, B. A Priori Error Estimates for the Finite Element Approximation of Westervelt’s Quasi-linear Acoustic Wave Equation. SIAM Journal on Numerical Analysis vol. 57 1897–1918 (2019) – 10.1137/19m1240873
- Meliani, M. & Nikolić, V. Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis. Applied Numerical Mathematics vol. 198 94–111 (2024) – 10.1016/j.apnum.2023.12.001
- Kaltenbacher, B. & Lehner, P. A first order in time wave equation modeling nonlinear acoustics. Journal of Mathematical Analysis and Applications vol. 543 128933 (2025) – 10.1016/j.jmaa.2024.128933
- Kaltenbacher, B., Nikolic, V. & Thalhammer, M. Efficient time integration methods based on operator splitting and application to the Westervelt equation. IMA Journal of Numerical Analysis vol. 35 1092–1124 (2014) – 10.1093/imanum/dru029
- Kaltenbacher B., Convergence of Implicit Runge‐Kutta Time Discretisation Methods for Fundamental Models in Nonlinear Acoustics. Journal of Applied & Numerical Optimization (2021)
- Nikolić, V. Asymptotic-preserving finite element analysis of Westervelt-type wave equations. Analysis and Applications 1–29 (2024) doi:10.1142/s0219530524500404 – 10.1142/s0219530524500404
- Wloka, J. Partial Differential Equations. (1987) doi:10.1017/cbo9781139171755 – 10.1017/cbo9781139171755
- Evans, L. Partial Differential Equations. Graduate Studies in Mathematics (2010) doi:10.1090/gsm/019 – 10.1090/gsm/019
- Boyer F., Mathematical Tools for the Study of the Incompressible Navier‐Stokes Equations and Related Models (2012)
- Roubíček, T. Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics (Springer Basel, 2012). doi:10.1007/978-3-0348-0513-1 – 10.1007/978-3-0348-0513-1
- From Finite to Infinite Dimensional Dynamical Systems. NATO Science Series II: Mathematics, Physics and Chemistry (Springer Netherlands, 2001). doi:10.1007/978-94-010-0732-0 – 10.1007/978-94-010-0732-0
- Qin, Y. Analytic Inequalities and Their Applications in PDEs. Operator Theory: Advances and Applications (Springer International Publishing, 2017). doi:10.1007/978-3-319-00831-8 – 10.1007/978-3-319-00831-8
- Egger, H. Structure preserving approximation of dissipative evolution problems. Numerische Mathematik vol. 143 85–106 (2019) – 10.1007/s00211-019-01050-w
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. (2014) doi:10.1561/9781601987877 – 10.1561/9781601987877