Advertisement

The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1018–1044 | Cite as

Wave Decay on Manifolds with Bounded Ricci Tensor, and Related Estimates

  • Michael TaylorEmail author
Article
  • 108 Downloads

Abstract

We investigate energy decay for solutions to the wave equation \(\partial_{t}^{2}u+a(x)\partial_{t}u-\Delta u=0\), with damping coefficient a≥0, where Δ is the Laplace–Beltrami operator on a compact Riemannian manifold M. We make a weak regularity hypothesis on the metric tensor of M, though one that guarantees the unique existence of the geodesic flow. We then establish exponential energy decay under the natural hypothesis that all sufficiently long geodesics pass through a region where a(x)≥a 0>0, extending the scope of previous work done in the setting of a smooth metric tensor.

Keywords

Damped wave equation Energy decay Microlocal regularity Ricci tensor 

Mathematics Subject Classification (2010)

35L10 35R01 

Notes

Acknowledgements

Thanks to anonymous referees for valuable suggestions to improve this paper.

References

  1. 1.
    Anderson, M., Katsuda, A., Kurylev, Y., Lassas, M., Taylor, M.: Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem. Invent. Math. 158, 261–321 (2004) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order. J. Math. Pures Appl. 36, 235–249 (1957) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bamberger, A., Rauch, J., Taylor, M.: The formation of harmonics on stringed instruments. Arch. Ration. Mech. Anal. 79, 267–290 (1982) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bony, J.-M.: Calcul symbolique et propagation des singularitiés pour les équations aux dérivées non linéaires. Ann. Sci. Éc. Norm. Super. 14, 209–246 (1981) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Cheeger, J.: Degeneration of Riemannian Manifolds with Ricci Curvature Bounds. Scuola Norm. Sup. Publ., Classe de Scienze, Pisa (2001) Google Scholar
  7. 7.
    Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Christianson, H.: Semiclassical non-concentration near hyperbolic orbits. J. Funct. Anal. 246, 145–195 (2007). Corrigendum, J. Funct. Anal. 258, 1060–1065 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cordes, H.: Über die eindeutige Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Acad. Wiss. Göttingen Math.-Phys. Kl IIa(11), 239–258 (1956) MathSciNetGoogle Scholar
  10. 10.
    De Hoop, M., Uhlmann, G., Vasy, A.: Diffraction from conormal singularities. Preprint (2012) Google Scholar
  11. 11.
    DeTurck, D., Kazdan, J.: Some regularity theorems in Riemannian geometry. Ann. Sci. Éc. Norm. Super. 14, 249–260 (1981) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Hörmander, L.: On the existence and the regularity of solutions of linear pseudo-differential equations. Enseign. Math. 17, 99–163 (1971) zbMATHGoogle Scholar
  13. 13.
    Koch, H., Tataru, D.: On the spectrum of hyperbolic semigroups. Commun. Partial Differ. Equ. 20, 901–937 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Lebeau, G.: Équation des ondes amorties. In: Algebraic and Geometric Methods in Mathematical Physics. Math. Phys. Stud., vol. 19, pp. 73–109. Kluwer, Dordrecht (1996) CrossRefGoogle Scholar
  15. 15.
    Lions, J., Magenes, E.: Problèmes aux Limites Non-homogènes et Applications, vol. 1. Dunod, Paris (1968) zbMATHGoogle Scholar
  16. 16.
    Marzuola, J., Metcalfe, J., Tataru, D.: Wave packet parametrices for evolutions governed by PDO’s with rough symbols. J. Funct. Anal. 255, 1497–1553 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Metcalfe, J., Tataru, D.: Global parametrices and dispersive estimates for variable coefficient wave equations. Math. Ann. 353, 1183–1237 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ralston, J.: Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22, 807–824 (1969) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Rauch, J., Taylor, M.: Penetration into shadow regions and unique continuation properties in hyperbolic mixed problems. Indiana Univ. Math. J. 22, 277–285 (1972) CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Rauch, J., Taylor, M.: Exponential decay of solutions to symmetric hyperbolic systems on bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974) CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Rauch, J., Taylor, M.: Decay of solutions to nondissipative hyperbolic systems on compact manifolds. Commun. Pure Appl. Math. 28, 501–523 (1975) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Smith, H.: A parametrix construction for wave equations with C 1,1 coefficients. Ann. Inst. Fourier (Grenoble) 48, 879–916 (1998) CrossRefGoogle Scholar
  23. 23.
    Smith, H.: Strichartz and nullform estimates for metrics of bounded curvature. Unpubl. Manuscript (1998) Google Scholar
  24. 24.
    Tataru, D.: Unique continuation for solutions to PDEs: between Hörmander’s theorem and Holmgren’s theorem. Commun. Partial Differ. Equ. 20, 855–884 (1995) CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Taylor, M.: Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Boston (1991) CrossRefzbMATHGoogle Scholar
  26. 26.
    Taylor, M.: Partial Differential Equations, vols. 1–3. Springer, New York (1996) (2nd edn. 2011) CrossRefGoogle Scholar
  27. 27.
    Taylor, M.: Tools for PDE. Math. Surv. and Monogr., vol. 81. Amer. Math. Soc., Providence (2000) zbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.University of North CarolinaChapel HillUSA

Personalised recommendations