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Klainerman, Sergiu; Selberg, Sigmund

Bilinear estimates and applications to nonlinear wave equations. (English) Zbl 1146.35389

Commun. Contemp. Math. 4, No. 2, 223-295 (2002).
Summary: We undertake a systematic review of results proved by S. Klainerman and M. Machedon [Duke Math. J. 81, No. 1, 99–133 (1995; Zbl 0909.35094); Duke Math. J. 87, No. 3, 553–589 (1997; Zbl 0878.35075); Differ. Integral Equ. 10, No. 6, 1019–1030 (1997; Zbl 0940.35011)], Klainerman and S. Selberg [Commun. Partial Differ. Equations 22, No. 5-6, 901–918 (1997; Zbl 0884.35102)], and Klainerman and D. Tataru [J. Am. Math. Soc. 12, No. 1, 93–116 (1999; Zbl 0924.58010)] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

Cited in 3 Reviews
Cited in 60 Documents

MSC:

35L70 Second-order nonlinear hyperbolic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35L30 Initial value problems for higher-order hyperbolic equations

Citations:

Zbl 0909.35094; Zbl 0878.35075; Zbl 0940.35011; Zbl 0884.35102; Zbl 0924.58010
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\textit{S. Klainerman} and \textit{S. Selberg}, Commun. Contemp. Math. 4, No. 2, 223--295 (2002; Zbl 1146.35389)
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