Advances in Differential Equations

Null-form estimates and nonlinear waves

Joachim Krieger

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We present bilinear and trilinear as well as quadrilinear null-form estimates arising in connection with the wave-maps problem. These estimates fit into the program of S. Klainerman begun in papers such as [4] and [1]. While the latter used the framework of $X^{s,b}$ spaces, our estimates involve the Banach spaces introduced by D. Tataru in [13] and further developed by T. Tao in [12]. In this paper we attempt to give a somewhat systematic account of the basic properties of (a certain brand of) these spaces in $n=3$ space dimensions. In particular, we solve some cases of the "division and summation problem" for semilinear wave equations whose nonlinearity has null-form structure.

Article information

Adv. Differential Equations, Volume 8, Number 10 (2003), 1193-1236.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B45: A priori estimates 35L75: Nonlinear higher-order hyperbolic equations


Krieger, Joachim. Null-form estimates and nonlinear waves. Adv. Differential Equations 8 (2003), no. 10, 1193--1236.

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