Duke Mathematical Journal

Improved local well-posedness for quasilinear wave equations in dimension three

S. Klainerman and I. Rodnianski

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We improve recent results of H. Bahouri and J.-Y. Chemin and of D. Tataru concerning local well-posedness theory for quasilinear wave equations. Our approach is based on the proof of the Strichartz estimates using a combination of geometric methods and harmonic analysis. The geometric component relies on and takes advantage of the nonlinear structure of the equation.

Article information

Duke Math. J. Volume 117, Number 1 (2003), 1-124.

First available in Project Euclid: 26 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35L15: Initial value problems for second-order hyperbolic equations 58J45: Hyperbolic equations [See also 35Lxx]


Klainerman, S.; Rodnianski, I. Improved local well-posedness for quasilinear wave equations in dimension three. Duke Math. J. 117 (2003), no. 1, 1--124. doi:10.1215/S0012-7094-03-11711-1. http://projecteuclid.org/euclid.dmj/1085598339.

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  • H. Bahouri and J.-Y. Chemin, Équations d'ondes quasilinéaires et effet dispersif, Internat. Math. Res. Notices 1999, 1141--1178.
  • --. --. --. --., Équations d'ondes quasilinéaires et estimation de Strichartz, Amer. J. Math. 121 (1999), 1337--1377.
  • D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton Math. Ser. 41, Princeton Univ. Press, Princeton, 1993.
  • D. Hoffman and J. Spruck, Sobolev and isoperimetric inequalities for Riemannian submanifolds, Comm. Pure Appl. Math. 27 (1974), 715--727.
  • L. V. Kapitanskiĭ, Estimates for norms in Besov and Lizorkin-Triebel spaces for solutions of second-order linear hyperbolic equations, J. Soviet Math. 56 (1991), 2348--2389.
  • S. Klainerman, A commuting vectorfields approach to Strichartz-type inequalities and applications to quasi-linear wave equations, Internat. Math. Res. Notices 2001, 221--274.
  • H. Lindblad, Counterexamples to local existence for semi-linear wave equations, Amer. J. Math. 118 (1996), 1--16.
  • G. Mockenhaupt, A. Seeger, and C. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), 65--130.
  • G. Ponce and T. Sideris, Local regularity of nonlinear wave equations in three space dimensions, Comm. Partial Differential Equations 18 (1993), 169--177.
  • L. Simon, Lectures on Geometric Measure Theory, Proc. Centre Math. Anal. Austral. Nat. Univ. 3, Australian National Univ., Canberra, 1983.
  • H. F. Smith, A parametrix construction for wave equations with $C^1,1$ coefficients, Ann. Inst. Fourier (Grenoble) 48 (1998), 797--835.
  • H. F. Smith and C. D. Sogge, On Strichartz and eigenfunction estimates for low regularity metrics, Math. Res. Lett. 1 (1994), 729--737.
  • H. F. Smith and D. Tataru, Sharp counterexamples for Strichartz estimates for low regularity metrics, Math. Res. Lett. 9 (2002), 199--204. \CMP1 909 638
  • D. Tataru, Strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation, Amer. J. Math. 122 (2000), 349--376.
  • --. --. --. --., Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients, II, Amer. J. Math. 123 (2001), 385--423.
  • --. --. --. --., Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients, III, J. Amer. Math. Soc. 15 (2002), 419--442.