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Large-Time Asymptotic Behaviors for Linear Blackstock’s Model of Thermoviscous Flow

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Abstract

In the classical theory of acoustic waves, Blackstock’s model was proposed in 1963 to characterize the propagation of sound in thermoviscous fluids. In this paper, we investigate large-time asymptotic behaviors of the linear Cauchy problem for general Blackstock’s model (that is, without Becker’s assumption on monatomic perfect gases). We derive first and second asymptotic profiles of solution as \(t\gg 1\) by applying refined WKB analysis and Fourier analysis. Our results not only improve optimal estimates in Chen et al. (Indiana Univ Math J 72(2), 2023) for lower dimensional cases, but also illustrate the optimal leading term and novel second profiles of solution with additional weighted \(L^1\) data.

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References

  1. Becker, R.: Stoßbwelle und detonation. Z. Phys. 8 (1922), 321–362 (in German), [English transl.: Impact waves and detonation, Part I, N.A.C.A. Technical Memo. No. 505 (N.A.C.A., Washington, DC, 1929); Part II, N.A.C.A. Technical Memo. No. 506 (N.A.C.A., Washington, DC, 1929)]

  2. Blackstock, D.T.: Approximate equations governing finite-amplitude sound in thermoviscous fluids. GD/E report GD-1463-52, General Dynamics Coporation (1963)

  3. Brunnhuber, R.: Well-posedness and long-time behavior of solutions for the Blackstock-Crighton equation. PhD thesis, Alpen-Adria-Universität Klagenfurt (2015)

  4. Brunnhuber, R.: Well-posedness and exponential decay of solutions for the Blackstock-Crighton-Kuznetsov equation. J. Math. Anal. Appl. 433(2), 1037–1054 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brunnhuber, R., Jordan, P.: On the reduction of Blackstock’s model of thermoviscous compressible flow via Becker’s assumption. Int. J. Non-Linear Mech. 78, 131–132 (2016)

    Article  Google Scholar 

  6. Brunnhuber, R., Kaltenbacher, B.: Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discret. Contin. Dyn. Syst. 34(11), 4515–4535 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brunnhuber, R., Meyer, S.: Optimal regularity and exponential stability for the Blackstock-Crighton equation in \(L_p\) spaces with Dirichlet and Neumann boundary conditions. J. Evol. Equ. 16(4), 945–981 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Celik, A., Kyed, M.: Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term. J. Math. Fluid Mech. 21(3), 1–12 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, W., Ikehata, R.: The Cauchy problem for the Moore-Gibson-Thompson equation in the dissipative case. J. Differ. Equ. 292, 176–219 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen, W., Ikehata, R., Palmieri, A.: Asymptotic behaviors for Blackstock’s model of thermoviscous flow. Indiana Univ. Math. J. 72(2) (2023), accepted

  11. Coulouvrat, F.: On the equations of nonlinear acoustics. J. de Acoustique 5, 321–359 (1992)

    Google Scholar 

  12. Ebert, M.R., Reissig, M.: Methods for Partial Differential Equations. Birkhäuser, Basel (2018)

    Book  MATH  Google Scholar 

  13. Gambera, L.R., Lizama, C., Prokopczyk, A.: Well-posedness for the abstract Blackstock-Crighton-Westervelt equation. J. Evol. Equ. 21(1), 313–337 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hamilton, M.F., Blackstock, D.T.: Nonlinear Acoustics. Academic Press, San Diego (1998)

    Google Scholar 

  15. Hayes, W.D.: Gas Dynamic Discontinuities. Princeton University Press, Princeton (1960)

    Book  Google Scholar 

  16. Ikehata, R.: Asymptotic profiles for wave equations with strong damping. J. Differ. Equ. 257(6), 2159–2177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ikehata, R.: \(L^2\)-blowup estimates of the plate equation. Preprint (2022)

  18. Ikehata, R.: \(L^2\)-blowup estimates of the wave equation and its application to local energy decay. J. Hyperbolic Differ. Equ. (2023), accepted

  19. Ikehata, R., Onodera, M.: Remarks on large time behavior of the \(L^2\)-norm of solutions to strongly damped wave equations. Differ. Integral Equ. 30(7–8), 505–520 (2017)

    MATH  Google Scholar 

  20. Kaltenbacher, B., Thalhammer, M.: Fundamental models in nonlinear acoustics part I. Analytical comparison. Math. Models Methods Appl. Sci. 28(12), 2403–2455 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, Y., Qin, X., Zhang, S.: Global existence and estimates for Blackstock’s model of thermoviscous flow with second sound phenomena. J. Differ. Equ. 324, 76–101 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Morduchow, M., Libby, P.A.: On a complete solution of the one-dimensional flow equations of a viscous, heat-conducting, compressible gas. J. Aeronaut. Sci. 16, 674–684 (1949)

    Article  MathSciNet  Google Scholar 

  23. Pierce, A.D.: Acoustics: An Introduction to its Physical Principles and Applications. Acoustical Society of America, Woodbury (1989)

    Google Scholar 

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Acknowledgements

The first author (Wenhui Chen) was partially supported by the National Natural Science Foundation of China under Grant No. 12171317, and Guangdong Basic and Applied Basic Research Foundation No. 2023A1515012044. The second author (Hirhoshi Takeda) is supported in part by the Grant-in-Aid for Scientific Research (C), No. 19K03596, from Japan Society for the Promotion of Science. The authors thank Ryo Ikehata (Hiroshima University) and Ya-guang Wang (Shanghai Jiao Tong University) for the suggestions in the preparation of the paper. The authors thank the anonymous referees for carefully reading the paper and giving some useful suggestions.

Funding

The first author (Wenhui Chen) was partially supported by the National Natural Science Foundation of China under Grant No. 12171317, and Guangdong Basic and Applied Basic Research Foundation No. 2023A1515012044. The second author (Hirhoshi Takeda) is supported in part by the Grant-in-Aid for Scientific Research (C), No. 19K03596, from Japan Society for the Promotion of Science.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Wenhui Chen

  2. Department of Intelligent Mechanical Engineering, Faculty of Engineering, Fukuoka Institute of Technology, Fukuoka, 811-0295, Japan

    Hiroshi Takeda

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  1. Wenhui Chen
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  2. Hiroshi Takeda
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WC wrote the main manuscript test; WC and HT did the computations of all the paper; All authors reviewed the manuscript.

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Correspondence to Wenhui Chen.

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Chen, W., Takeda, H. Large-Time Asymptotic Behaviors for Linear Blackstock’s Model of Thermoviscous Flow. Appl Math Optim 88, 21 (2023). https://doi.org/10.1007/s00245-023-10003-7

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