The solid-fluid transmission problem

Eptaminitakis, Nikolas; Stefanov, Plamen

doi:10.1090/tran/9016

The solid-fluid transmission problem
HTML articles powered by AMS MathViewer

by Nikolas Eptaminitakis and Plamen Stefanov

Trans. Amer. Math. Soc. 377 (2024), 2583-2633

DOI: https://doi.org/10.1090/tran/9016

Published electronically: February 9, 2024

HTML | PDF | Request permission

Abstract:

We study microlocally the transmission problem at the interface between an isotropic linear elastic solid and a linear inviscid fluid. We set up a system of evolution equations describing the particle displacement and velocity in the solid, and pressure and velocity in the fluid, coupled by suitable transmission conditions at the interface. We show well-posedness for the coupled system and study the problem microlocally, constructing a parametrix for it using geometric optics. This construction describes the reflected and transmitted waves, including mode converted ones, related to incoming waves from either side. We also study formation of surface Scholte waves. Finally, we prove that under suitable assumptions, we can recover the s- and the p-speeds, as well as the speed of the liquid, from boundary measurements.

References

James H. Ansell, The roots of the Stoneley wave equation for solid-liquid interfaces, Pure Appl. Geophys. 94 (1972), no. 1, 172–188.
Keiiti Aki and Paul Richards, Quantitative seismology, 2nd ed., University Science Books, Sausalito, Calif., 2002.
Gang Bao, Yixian Gao, and Peijun Li, Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal. 229 (2018), no. 2, 835–884. MR 3803777, DOI 10.1007/s00205-018-1228-2
Guillaume Bal, Joseph B. Keller, George Papanicolaou, and Leonid Ryzhik, Transport theory for acoustic waves with reflection and transmission at interfaces, Wave Motion 30 (1999), no. 4, 303–327. MR 1711949, DOI 10.1016/S0165-2125(99)00018-9
Sombuddha Bhattacharyya, Maarten V. de Hoop, Vitaly Katsnelson, and Gunther Uhlmann, Recovery of piecewise smooth density and Lamé parameters from high frequency exterior Cauchy data, SIAM J. Imaging Sci. 15 (2022), no. 4, 1910–1943. MR 4512265, DOI 10.1137/22M1480951
Sombuddha Bhattacharyya, Maarten V. de Hoop, Vitaly Katsnelson, and Gunther Uhlmann, Recovery of wave speeds and density of mass across a heterogeneous smooth interface from acoustic and elastic wave reflection operators, GEM Int. J. Geomath. 13 (2022), no. 1, Paper No. 9, 46. MR 4423644, DOI 10.1007/s13137-022-00199-1
Peter Caday, Maarten V. de Hoop, Vitaly Katsnelson, and Gunther Uhlmann, Recovery of discontinuous Lamé parameters from exterior Cauchy data, Comm. Partial Differential Equations 46 (2021), no. 4, 680–715. MR 4260458, DOI 10.1080/03605302.2020.1857399
Yanli Cui and Fenglong Qu, Identification of the interface between acoustic and elastic waves from internal measurements, J. Inverse Ill-Posed Probl. 28 (2020), no. 3, 313–322. MR 4104321, DOI 10.1515/jiip-2018-0101
Nils Dencker, On the propagation of polarization sets for systems of real principal type, J. Functional Analysis 46 (1982), no. 3, 351–372. MR 661876, DOI 10.1016/0022-1236(82)90051-9
Maarten V. de Hoop, Sean Holman, and Ha Pham, On the system of elastic-gravitational equations describing the oscillations of the earth, 2017.
Avron Douglis and Louis Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math. 8 (1955), 503–538. MR 75417, DOI 10.1002/cpa.3160080406
F. A. Dahlen and Jeroen Tromp, Theoretical global seismology, Princeton University Press, Princeton, 1998.
J. Y. Guo, P. M. Mathews, Z. X. Zhang, and J. S. Ning, Impact of inner core rotation on outer core flow: the role of outer core viscosity, Geophys. J. Int. 159 (2004), no. 1, 372–389.
Alain Grigis and Johannes Sjöstrand, Microlocal analysis for differential operators, London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. An introduction. MR 1269107, DOI 10.1017/CBO9780511721441
Sönke Hansen, Propagation of polarization in transmission problems, Pure Appl. Anal. 4 (2022), no. 1, 153–190. MR 4419371, DOI 10.2140/paa.2022.4.153
Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1892228
C. J. Luke and P. A. Martin, Fluid-solid interaction: acoustic scattering by a smooth elastic obstacle, SIAM J. Appl. Math. 55 (1995), no. 4, 904–922. MR 1341532, DOI 10.1137/S0036139993259027
Peijun Li and Lei Zhang, Analysis of transient acoustic scattering by an elastic obstacle, Commun. Math. Sci. 17 (2019), no. 6, 1671–1698. MR 4052755, DOI 10.4310/CMS.2019.v17.n6.a8
William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
Jerrold E. Marsden and Thomas J. R. Hughes, Mathematical foundations of elasticity, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1983 original. MR 1262126
Lizabeth V. Rachele, Boundary determination for an inverse problem in elastodynamics, Comm. Partial Differential Equations 25 (2000), no. 11-12, 1951–1996. MR 1789918, DOI 10.1080/03605300008821575
J. G. Scholte, The range of existence of Rayleigh and Stoneley waves, Geophys. J. Int. 5 (1947), 120–126.
E. Strick and A. S. Ginzbarg, Stoneley-wave velocities for a fluid-solid interface, Bull. Seismological Soc. Amer. 46 (1956), no. 4, 281–292.
Robert E. Sheriff and Lloyd Geldart, Exploration seismology, 2nd ed., Cambridge University Press, Cambridge, New York, 1995.
Plamen Stefanov, Gunther Uhlmann, and Andras Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc. 29 (2016), no. 2, 299–332. MR 3454376, DOI 10.1090/jams/846
Plamen Stefanov, Gunther Uhlmann, and Andras Vasy, Local recovery of the compressional and shear speeds from the hyperbolic DN map, Inverse Problems 34 (2018), no. 1, 014003, 13. MR 3742360, DOI 10.1088/1361-6420/aa9833
Plamen Stefanov, Gunther Uhlmann, and András Vasy, The transmission problem in linear isotropic elasticity, Pure Appl. Anal. 3 (2021), no. 1, 109–161. MR 4265359, DOI 10.2140/paa.2021.3.109
P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body, Duke Math. J. 78 (1995), no. 3, 677–714. MR 1334206, DOI 10.1215/S0012-7094-95-07825-9
Michael E. Taylor, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon, Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977) Lect. Notes Pure Appl. Math., vol. 48, Dekker, New York, 1979, pp. 273–291. MR 535598
Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, No. 34, Princeton University Press, Princeton, NJ, 1981. MR 618463, DOI 10.1515/9781400886104
Mark Williams, Transmission across a moving interface: necessary and sufficient conditions for $(L^2)$ well-posedness, Indiana Univ. Math. J. 41 (1992), no. 2, 303–338. MR 1183346, DOI 10.1512/iumj.1992.41.41018
Kazuhiro Yamamoto, Reflective and refractive phenomena of tangential elastic waves to the boundary in two solids as a propagation of singularities, J. Math. Pures Appl. (9) 92 (2009), no. 2, 188–208 (English, with English and French summaries). MR 2542583, DOI 10.1016/j.matpur.2009.05.001