Improved local well-posedness for quasilinear wave equations in dimension three. (English) Zbl 1031.35091

Klainerman, S.; Rodnianski, I.

Duke Math. J. 117, No. 1, 1-124 (2003).

This paper is devoted to the initial value problem for quasilinear wave equations of the form \[ \begin{gathered} \partial^2_t\varphi- g^{ij}(\varphi) \partial_i\partial_j\varphi= N(\varphi,\partial\varphi),\\ \varphi|_{t=0}= \varphi_0,\quad \partial_t\varphi|_{t=0}= \varphi_1.\end{gathered}\tag{1} \] The authors improve recent results of H. Bahouri and J. Y. Chernin and of D. Tataru concerning local well-posedness for (1). Their approach is based on the proof of the Strichartz estimates using a combination of geometric methods and harmonic analysis. The authors use in an essential way the fact that the coefficients \(g^{ij}(\varphi)\) in (1) themselves verify an equation of the type \(\square_g g_{ij}= N\) with \(N\) depending only on \(\varphi\) and \(\partial\varphi\). Here \(\square\) denotes standard linear wave operator.

Reviewer: Messoud Efendiev (Berlin)

MSC:

35L15	Initial value problems for second-order hyperbolic equations
35L70	Second-order nonlinear hyperbolic equations
58J45	Hyperbolic equations on manifolds
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

Strichartz estimates

References:

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