Duke Mathematical Journal

A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations

Hans Lindblad

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Duke Math. J., Volume 72, Number 2 (1993), 503-539.

First available in Project Euclid: 20 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35L15: Initial value problems for second-order hyperbolic equations 35L67: Shocks and singularities [See also 58Kxx, 76L05]


Lindblad, Hans. A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations. Duke Math. J. 72 (1993), no. 2, 503--539. doi:10.1215/S0012-7094-93-07219-5. https://projecteuclid.org/euclid.dmj/1077289430

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  • [1] M. Beals and M. Bezard, Low-regularity local solutions for field equations, preprint, 1992.
  • [2] P. Brenner, On $L\sbp-L\sbp\sp\prime $ estimates for the wave-equation, Math. Z. 145 (1975), no. 3, 251–254.
  • [3] C. Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336.
  • [4] M. Grillakis, Regularity for the wave equation with a critical nonlinearity, Comm. Pure Appl. Math. 45 (1992), no. 6, 749–774.
  • [5] J. Harmse, On Lebesgue space estimates for the wave equation, Indiana Univ. Math. J. 39 (1990), no. 1, 229–248.
  • [6] L. Hörmander, Estimates for translation invariant operators in $L\spp$ spaces, Acta Math. 104 (1960), 93–140.
  • [7] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, Pseudodifferential operators (Oberwolfach, 1986), Lecture Notes in Math., vol. 1256, Springer, Berlin, 1987, pp. 214–280.
  • [8] L. V. Kapitanskiĭ, Some generalizations of the Strichartz-Brenner inequality, Algebra i Analiz 1 (1989), no. 3, 127–159.
  • [9] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907.
  • [10] C. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math. 46 (1993), no. 4, 527–620.
  • [11] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, preprint, 1991.
  • [12] H. Lindblad, Blow-up for solutions of $\square u=\vert u\vert \sp p$ with small initial data, Comm. Partial Differential Equations 15 (1990), no. 6, 757–821.
  • [13] H. Pecher, $L\spp$-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen. I, Math. Z. 150 (1976), no. 2, 159–183.
  • [14] G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, preprint, 1991.
  • [15] J. Rauch, Explosion for some semilinear wave equations, J. Differential Equations 74 (1988), no. 1, 29–33.
  • [16] J. Ralston, Gaussian beams and the propagation of singularities, Studies in partial differential equations, MAA Stud. Math., vol. 23, Math. Assoc. America, Washington, DC, 1982, pp. 206–248.
  • [17] I. Segal, Space-time decay for solutions of wave equations, Advances in Math. 22 (1976), no. 3, 305–311.
  • [18] J. Shatah and Shadi A. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, preprint, 1992.
  • [19] C. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993.
  • [20] E. M. Stein, Oscillatory integrals in Fourier analysis, Beijing lectures in harmonic analysis (Beijing, 1984), Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355.
  • [21] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.
  • [22] R. S. Strichartz, Convolutions with kernels having singularities on a sphere, Trans. Amer. Math. Soc. 148 (1970), 461–471.
  • [23] R. S. Strichartz, A priori estimates for the wave equation and some applications, J. Functional Analysis 5 (1970), 218–235.