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Smoothing Estimates for the Wave Equation and Applications

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Proceedings of the International Congress of Mathematicians

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Abstract

The purpose of this paper is to go over some recent results in analysis and partial differential equations that are related to regularity properties of solutions of the wave equation

$$ \left\{ {_{u(0,x) = f(x),{\partial _t}u(0,x) = g(x).}^{\square u(t,x) = 0}} \right.$$
(1)

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References

  1. M. Beals,Lp boundedness of Fourier integrals, Mem. Amer. Math. Soc.264 (1982).

    Google Scholar 

  2. J. Bourgain,Averages in the plane over convex curves and maximal operators, J. Analyse Math.47 (1986), 69 – 85.

    Article  MathSciNet  Google Scholar 

  3. ______,Besicovitch type maximal operators and applications to Fourier analysis, Geom. Functional Anal.1 (1991), 69 – 85.

    MathSciNet  MATH  Google Scholar 

  4. P. Brenner,On Lp— LP′estimates for the wave equation, Math. Z.145 (1975), 251 – 254.

    Article  MathSciNet  Google Scholar 

  5. A. Carbery,The boundedness of the maximal Bochner-Riesz opearator on L4(R2), Duke Math. J. (1983), 409 – 416.

    Google Scholar 

  6. L. Carleson and P. Sjolin,Oscillatory integrals and a multiplier problem for the disk, Studia Math.44 (1972), 287 – 299.

    Article  MathSciNet  Google Scholar 

  7. F. M. Christ and C. D. Sogge,The weak type L1convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math.94 (1988), 421 – 451.

    MATH  Google Scholar 

  8. A. Cordoba,A note on Bochner-Riesz operators, Duke Math. J.46 (1979), 505 – 511.

    Article  MathSciNet  Google Scholar 

  9. K. J. Falconer, The geometry of fractal sets, Cambridge Univ. Press, Cambridge, 1985.

    Book  Google Scholar 

  10. C. Fefferman,The multiplier problem for the ball, Ann. of Math. (2)94 (1971), 330 – 336.

    Article  MathSciNet  Google Scholar 

  11. ______,A note on spherical summation multipliers, Israel J. Math.15 (1973), 44 – 52.

    Article  MathSciNet  Google Scholar 

  12. J. Ginibre and G. Velo,Conformal invariance and time decay for nonlinear wave equations, II, Ann. Inst. H. Poincare Phys. Theor.47 (1987), 263 – 276.

    MathSciNet  MATH  Google Scholar 

  13. _______,Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys.123 (1989), 535 – 573.

    Article  MathSciNet  Google Scholar 

  14. D. Grieser,Lpbounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries, thesis, UCLA (1992).

    Google Scholar 

  15. M. G. Grillakis,Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math.45 (1992), 749 – 774.

    Article  MathSciNet  Google Scholar 

  16. L. Hormander,The spectral function of an elliptic operator, Acta Math.121 (1968), 193 – 218.

    Article  MathSciNet  Google Scholar 

  17. _______,Fourier integral operators I, Acta Math.127 (1971), 79 – 183.

    Article  MathSciNet  Google Scholar 

  18. ______ ,Oscillatory integrals and multipliers on FLP, Ark. Mat.11 (1971), 1 – 11.

    Article  Google Scholar 

  19. A. Iosevich,Maximal operators associated to the families of flat curves in the plane, Duke Math. J.76 (1994), 633 – 644.

    Article  MathSciNet  Google Scholar 

  20. V. Ivrii, Precise spectral asymptotics for elliptic operators, Lecture Notes in Math. 1100, Springer-Verlag, Berlin and New York, 1984.

    Google Scholar 

  21. F. John,Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math.28 (1979), 235 – 265.

    Article  MathSciNet  Google Scholar 

  22. L. Kapitanski,Some generalizations of the Strichartz-Brenner inequality, Leningrad Math. J.1 (1990), 693 – 726.

    MathSciNet  MATH  Google Scholar 

  23. ______, Weak and yet weaker solutions of semilinear wave equations, Brown Univ. preprint, Providence, RI.

    Google Scholar 

  24. T. Kato,On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Math., vol. 8, Academic Press, New York and San Diego, 1983, pp. 93 – 128.

    Google Scholar 

  25. J. B. Keller,On solutions of nonlinear wave equations, Comm. Pure Appl. Math.10 (1957), 523 – 530.

    Article  MathSciNet  Google Scholar 

  26. S. Klainerman and M. Machedon,Space-time estimates for null forms and the local existence theorem, Comm. Pure and Appl. Math.46 (1993), 1221 – 1268.

    Article  MathSciNet  Google Scholar 

  27. H. Lindblad,Blow-up for solutions of \3u = \u\p with small initial data, Comm. Partial Differential Equations15 (1990), 757 – 821.

    Article  MathSciNet  Google Scholar 

  28. H. Lindblad and C. D. Sogge,On existence and scattering with minimal regularity for semilinear wave equations, J. Funct. Anal., to appear.

    Google Scholar 

  29. ______, forthcoming.

    Google Scholar 

  30. W. Littman,The wave operator and Lp-norms, J. Math. Mech. 12 (1963), 55 – 68.

    MathSciNet  MATH  Google Scholar 

  31. A. Miyachi,On some estimates for the wave operator in Lpand Hp, J. Fac. Sci. Univ. Tokyo Sect. IA Math.27 (1980), 331 – 354.

    MathSciNet  MATH  Google Scholar 

  32. G. Mockenhaupt, A. Seeger, and C. D. Sogge,Wave front sets, local smoothing and Bourgain’s circular maximal theorem, Ann. Math. (2)136 (1992), 207 – 218.

    Article  MathSciNet  Google Scholar 

  33. ______,Local smoothing of Fourier integrals and Carles on-Sjolin estimates, J. Amer. Math. Soc.6 (1993), 65 – 130.

    MathSciNet  MATH  Google Scholar 

  34. D. Müller and A. Seeger,Inequalities for spherically symmetric solutions of the wave equation, Math. Z., to appear.

    Google Scholar 

  35. H. Pecher,Nonlinear small data scattering for the wave and Klein-Gordon equations, Math. Z.185 (1984), 261 – 270.

    Article  MathSciNet  Google Scholar 

  36. J. Peral,Lpestimates for the wave equation, J. Funct. Anal.36 (1980), 114 – 145.

    Article  MathSciNet  Google Scholar 

  37. D. H. Phong and E. M. Stein,Hilbert integrals, singular integrals and Radon transforms I, Acta Math.157 (1986), 99 – 157.

    Article  MathSciNet  Google Scholar 

  38. A. Seeger and C. D. Sogge,Bounds for eigenfunctions of differential operators, Indiana Univ. Math. J.38 (1989), 669 – 682.

    MathSciNet  MATH  Google Scholar 

  39. A. Seeger, C. D. Sogge, and E. M. Stein,Regularity properties of Fourier integral operators, Ann. Math. (2)134 (1991), 231 – 251.

    Article  MathSciNet  Google Scholar 

  40. I. E. Segal,Space-time decay for solutions of wave equations, Adv. in Math.22 (1976), 304 – 311.

    Article  MathSciNet  Google Scholar 

  41. J. Shatah and M. Struwe,Regularity results for nonlinear wave equations, Ann. of Math. (2)138 (1993), 503 – 518.

    Article  MathSciNet  Google Scholar 

  42. H. Smith and C. D. Sogge,Lp regularity for the wave equation with strictly convex obstacles, Duke Math. J.73 (1994), 123 – 134.

    Article  Google Scholar 

  43. ______,On the critical semilinear wave equation outside convex obstacles, Amer. Math. Soc., to appear.

    Google Scholar 

  44. C. D. Sogge,Concerning the Lp norm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal.77 (1988), 123 – 134.

    Article  MathSciNet  Google Scholar 

  45. ______,Propagation of singularities and maximal functions in the plane, Invent. Math.104 (1991), 349 – 376.

    Article  MathSciNet  Google Scholar 

  46. ______, Fourier integrals in classical analysis, Cambridge Univ. Press, Cambridge, 1993.

    Book  Google Scholar 

  47. ______,Averages over hypersurfaces with one non-vanishing principal curvature, Fourier analysis and partial differential equations (J. Garcia-Cuerva, ed.), CRC Press, Boca Raton, FL, 1995, pp. 317 – 323.

    Google Scholar 

  48. E. M. Stein,Maximal functions: Spherical means, Proc. Nat. Acad. Sci.73 (1976), 2174 – 2175.

    Article  MathSciNet  Google Scholar 

  49. ______,Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, NJ, 1988, pp. 307 – 356.

    Google Scholar 

  50. ______, Harmonic analysis real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  51. R. Strichartz,A priori estimates for the wave equation and some applications, J. Funct. Anal.5 (1970), 218 – 235.

    Article  MathSciNet  Google Scholar 

  52. ______,Restriction of Fourier transform to quadratic surfaces and decay of solutions to the wave equation, Duke Math. J.44 (1977), 705 – 714.

    Article  MathSciNet  Google Scholar 

  53. P. Tomas,Restriction theorems for the Fourier transform, Proc. Sympos. Pure Math.35 (1979), 111 – 114.

    Google Scholar 

  54. T. Wolff,An improved bound for Kakeya type maximal functions, preprint.

    Google Scholar 

  55. Y. Zhou,Blow up of classical solutions to \3u = M1+a in three space dimensions, J. Partial Differential Equations Ser. A5 (1992), 21 – 32.

    Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, University of California, Los Angeles, CA, 90024, USA

    Christopher D. Sogge

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  1. Christopher D. Sogge
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Editors and Affiliations

  1. Département de Mathématiques, EPFL, 1015, Laussane, Switzerland

    S. D. Chatterji

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© 1995 Birkhaäser Verlag

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Sogge, C.D. (1995). Smoothing Estimates for the Wave Equation and Applications. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_82

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  • DOI: https://doi.org/10.1007/978-3-0348-9078-6_82

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9897-3

  • Online ISBN: 978-3-0348-9078-6

  • eBook Packages: Springer Book Archive

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