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.
S. Klainerman. I. Rodnianski. "Improved local well-posedness for quasilinear wave equations in dimension three." Duke Math. J. 117 (1) 1 - 124, 15 March 2003. https://doi.org/10.1215/S0012-7094-03-11711-1