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2014 | OriginalPaper | Buchkapitel

15. Inverse Problem for a Linearized Jordan–Moore–Gibson–Thompson Equation

verfasst von : Shitao Liu, Roberto Triggiani

Erschienen in: New Prospects in Direct, Inverse and Control Problems for Evolution Equations

Verlag: Springer International Publishing

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Abstract

We consider an inverse problem for the linearized Jordan–Moore–Gibson–Thompson equation, which is a third-order (in time) PDE in the original unknown u that arises in nonlinear acoustic waves modeling high-intensity ultrasound. Both canonical recovery problems are investigated: (i) uniqueness and (ii) stability, by use of just one boundary measurement. Our approach relies on the dynamical decomposition of the Jordan–Moore–Gibson–Thompson equation given in Marchand et al. (Math. Methods Appl. Sci. 35, 1896–1929, 2012), which identified 3 distinct models in the new variable z. By using now z-model ♯3, we weaken by two units the regularity requirements on the data of the original u-dynamics over our prior effort Liu and Triggiani (J. Inverse Ill-Posed Probl. 21, 825–869, 2013), which instead employed z-model ♯1.

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Beilina, L., Klibanov, M.: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems. Springer, New York (2012)CrossRefMATH

Bukhgeim, A., Cheng, J., Isakov. V., Yamamoto, M.: Uniqueness in determining damping coefficients in hyperbolic equations. In: Saburou Saitoh, Nakao Hayashi, Masahiro Yamamoto (eds.), Analytic Extension Formulas and Their Applications, pp. 27–46. Kluwer, Dordrecht (2001)CrossRef

Bukhgeim, A., Klibanov, M.: Global uniqueness of a class of multidimensional inverse problem. Sov. Math. Dokl. 24, 244–257 (1981)

Carleman, T.: Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles à deux variables independantes. Ark. Mat. Astr. Fys.2B, 1–9 (1939)MathSciNet

Ho, L. F.: Observabilite frontiere de l’equation des ondes. Comptes Rendus de l’Academie des Sciences de Paris 302, 443–446 (1986)MATH

Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin/ New York (1985)

Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Springer, Berlin/ New York (1985)

Isakov, V.: Inverse Problems for Partial Differential Equations, 1st edn. Springer, New York (1998)CrossRefMATH

Isakov, V.: Inverse Problems for Partial Differential Equations, 2nd edn. Springer, New York (2006)MATH

10.

Isakov, V., Yamamoto, M.: Carleman estimate with the Neumann boundary condition and its application to the observability inequality and inverse hyperbolic problems. Contemp. Math. 268, 191–225 (2000)CrossRefMathSciNet

11.

Isakov, V., Yamamoto, M.: Stability in a wave source problem by Dirichlet data on subboundary. J. Inverse Ill-Posed Probl. 11, 399–409 (2003)CrossRefMATHMathSciNet

12.

Jordan, P.M.: An analytic study of the Kuznetsov’s equation: diffusive solitons, shock formation, and solution bifurcation. Phys. Lett. A 326, 77–84 (2004)CrossRefMATHMathSciNet

13.

Jordan, P.M.: Nonlinear acoustic phenomena in viscous thermally relaxing fluids: shock bifurcation and the emergence of diffusive solitions (A) (Lecture). The 9th International Conference on Theoretical and Computational Acoustics (ICTCA 2009), Dresden, Germany. J. Acoust. Soc. Am. 124, 2491–2491 (2008)

14.

Kaltenbacher, B., Lasiecka, I.: Global existence and exponential decay rates for the Westervelt equation. DCDS Ser. S 2, 503–525 (2009)CrossRefMATHMathSciNet

15.

Kaltenbacher, B., Lasiecka, I.: Well-posedness of the Westervelt and the Kuznetsov equations with non homogeneous Neumann boundary conditions. DCDS Suppl., 763–773 (2011)

16.

Kaltenbacher, B., Lasiecka, I., Marchand, R.: Wellposedness and exponential decay rates for the Moore–Gibson–Thompson equation arising in high intensity ultrasound. Control Cybern. (2011)

17.

Kaltenbacher, B., Lasiecka, I., Veljovic, S.: Well-posedness and exponential decay of the Westervelt equation with inhomogeneous Dirichlet boundary data. Progress in Nonlinear Differential Equations and Their Applications, vol. 60. Springer, Basel (2011)

18.

Klibanov, M.: Inverse problems and Carleman estimates. Inverse Probl. 8, 575–596 (1992)CrossRefMATHMathSciNet

19.

Klibanov, M.: Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems. J. Inverse Ill-Posed Probl. 21(2), (2013)

20.

Klibanov, M., Timonov, A.: Carleman Estimates for Coefficient Inverse Problems and Numerical Applications. VSP, Utrecht (2004)CrossRefMATH

21.

Kuznetsov, V.P.: Equations of nonlinear acoustics. Sov. Phys. 16, 467–470 (1971)

22.

Lasiecka, I., Lions, J.L., Triggiani, R.: Non-homogeneous boundary value problems for second-order hyperbolic operators. J. Math. Pures Appl. 65, 149–192 (1986)MATHMathSciNet

23.

Lasiecka, I., Triggiani, R.: A cosine operator approach to modeling L ₂(0, T; L ₂(Ω)) boundary input hyperbolic equations. Appl. Math. Optim. 7, 35–83 (1981)CrossRefMATHMathSciNet

24.

Lasiecka, I., Triggiani, R.: Regularity of hyperbolic equations under L ₂(0, T; L ₂(Γ))-Dirichlet boundary terms. Appl. Math. Optim. 10, 275–286 (1983)CrossRefMATHMathSciNet

25.

Lasiecka, I., Triggiani, R.: Exact controllability of the wave equation with Neumann boundary control. Appl. Math. Optim. 19, 243–290 (1989)CrossRefMATHMathSciNet

26.

Lasiecka, I., Triggiani, R.: Sharp regularity theory for second-order hyperbolic equations of Neumann type Part I: L ₂ non-homogeneous data. Ann. Mat. Pura Appl. (IV) CLVII, 285–367 (1990)

27.

Lasiecka, I., Triggiani, R.: Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions II: General boundary data. J. Differ. Equ. 94, 112–164 (1991)CrossRefMATHMathSciNet

28.

Lasiecka, I., Triggiani, R.: Recent advances in regularity of second-order hyperbolic mixed problems and applications. Dynamics Reported, vol. 3, pp. 104–158. Springer, New York (1994)

29.

Lasiecka, I., Triggiani, R.: Carleman estimates and uniqueness for the system of strong coupled PDE’s of spherical shells. Special volume of Zeits. Angerwandte Math. Mech. vol. 76, pp.277–280. Akademie, Berlin (1996)

30.

Lasiecka, I., Triggiani, R.: Carleman estimates and exact controllability for a system of coupled, nonconservative second-order hyperbolic equations. Lect. Notes Pure Appl. Math. 188, 215–245 (1997)MathSciNet

31.

Lasiecka, I., Triggiani, R.: Exact boundary controllability of a first-order nonlinear hyperbolic equation with non-local in the integral term arising in epidemic modeling. In: Gilbert, R.P., Kajiwara, J., Xu, Y. (eds.) Direct and Inverse Problems of Mathematical Physics, pp. 363–398. ISAAC’97, The First International Congress of the International Society for Analysis, Its Applications and Computations. Kluwer (2000)

32.

Lasiecka, I., Triggiani, R.: Uniform stabilization of the wave equation with Dirichlet or Neumann-feedback control without geometrical conditions. Appl. Math. Optim. 25, 189–224 (1992)CrossRefMATHMathSciNet

33.

Lasiecka, I., Triggiani, R., Yao, P.F.: Exact controllability for second-order hyperbolic equations with variable coefficient-principal part and first-order terms. Nonlinear Anal. 30(1), 111–222 (1997)CrossRefMATHMathSciNet

34.

Lasiecka, I., Triggiani, R., Yao, P.F.: Inverse/observability estimates for second-order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235(1), 13–57 (1999)CrossRefMATHMathSciNet

35.

Lasiecka, I., Triggiani, R., Zhang, X.: Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot. Contemp. Math. 268, 227–325 (2000)

36.

Lavrentev, M.M., Romanov, V.G., Shishataskii, S.P.: Ill-Posed Problems of Mathematical Physics and Analysis, vol. 64. The American Mathematical Society, Providence (1986)

37.

Lions, J.L.: Controlabilite Exacte, Perturbations et Stabilisation de Systemes Distribues, vol. 1. Masson, Paris (1988)

38.

Liu, S.: Inverse problem for a structural acoustic interaction. Nonlinear Anal. 74, 2647–2662 (2011)CrossRefMATHMathSciNet

39.

Liu, S., Triggiani, R.: Global uniqueness and stability in determining the damping and potential coefficients of an inverse hyperbolic problem. Nonlinear Anal. Real World Appl. 12, 1562–1590 (2011)CrossRefMATHMathSciNet

40.

Liu, S., Triggiani, R.: Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Neumann B.C. through an additional Dirichlet boundary trace. SIAM J. Math. Anal. 43, 1631–1666 (2011)

41.

Liu, S., Triggiani, R.: Global uniqueness in determining electric potentials for a system of strongly coupled Schrödinger equations with magnetic potential terms. J. Inverse Ill-Posed Probl. 19, 223–254 (2011)CrossRefMATHMathSciNet

42.

Liu, S., Triggiani, R.: Recovering the damping coefficients for a system of coupled wave equations with Neumann BC: uniqueness and stability. Chin. Ann. Math. Ser. B 32, 669–698 (2011)CrossRefMATHMathSciNet

43.

Liu, S., Triggiani, R.: Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: global uniqueness. DCDS Supplement, 1001–1014 (2011)

44.

Liu, S., Triggiani, R.: Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with non-homogeneous Dirichlet B.C. through an additional localized Neumann boundary trace. Appl. Anal.91(8), 1551–1581 (2012)

45.

Liu, S., Triggiani, R.: Recovering damping and potential coefficients for an inverse non-homogeneous second-order hyperbolic problem via a localized Neumann boundary trace. Discrete Contin. Dyn. Syst. Ser. A 33(11–12), 5217–5252 (2013)MATH

46.

Liu, S., Triggiani, R.: Boundary control and boundary inverse theory for non-homogeneous second-order hyperbolic equations: a common Carleman estimates approach. HCDTE Lecture notes, AIMS Book Series on Applied Mathematics, vol. 6, pp. 227–343 (2013)

47.

Liu, S., Triggiani, R.: An inverse problem for a third order PDE arising in high-intensity ultrasound: global uniqueness and stability by one boundary measurement. J. Inverse Ill-Posed Probl. 21, 825–869 (2013)CrossRefMATHMathSciNet

48.

Marchand, R., McDevitt, T., R. 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. Math. Methods Appl. Sci. 35, 1896–1929 (2012)

49.

Mazya, V.G., Shaposhnikova, T.O.: Theory of Multipliers in Spaces of Differentiable Functions, vol. 23. Monographs and Studies in Mathematics, Pitman (1985)

50.

Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

51.

Tataru, D.: A-priori estimates of Carleman’s type in domains with boundary. J. Math. Pures. et Appl. 73, 355–387 (1994)MATHMathSciNet

52.

Tataru, D.: Boundary controllability for conservative PDE’s. Appl. Math. & Optimiz. 31, 257–295 (1995); Based on a Ph.D. dissertation, University of Virginia (1992)

53.

Tataru, D.: Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures Appl. 75, 367–408 (1996)MATHMathSciNet

54.

Tataru, D.: On the regularity of boundary traces for the wave equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26, 185–206 (1998)

55.

Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)MATH

56.

Triggiani, R.: Exact boundary controllability of L ₂(Ω) × H ⁻¹(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary and related problems. Appl. Math. Optim. 18(3), 241–277 (1988)CrossRefMATHMathSciNet

57.

Triggiani, R.: Wave equation on a bounded domain with boundary dissipation: an operator approach. J. Math. Anal. Appl. 137, 438–461 (1989)CrossRefMATHMathSciNet

58.

Triggiani, R., Yao, P.F.: Carleman estimates with no lower order terms for general Riemannian wave equations: global uniqueness and observability in one shot. Appl. Math. Optim. 46, 331–375 (2002)CrossRefMATHMathSciNet

59.

Yamamoto, M.: Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78, 65–98 (1999)CrossRefMATHMathSciNet

Titel: Inverse Problem for a Linearized Jordan–Moore–Gibson–Thompson Equation
verfasst von: Shitao Liu
Roberto Triggiani
Verlag: Springer International Publishing
Buch: New Prospects in Direct, Inverse and Control Problems for Evolution Equations
Print ISBN: 978-3-319-11405-7

Electronic ISBN: 978-3-319-11406-4

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-11406-4_15

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