The Blowup Mechanism of Small Data Solutions for the Quasilinear Wave Equations in Three Space Dimensions

For a class of three-dimensional quasilinear wave equations with small initial data, we give a complete asymptotic expansion of the lifespan of classical solutions, that is, we solve a conjecture posed by John and Hrmander. As an application of our result, we show that the solution of three-dimensional isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singularity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conjecture of Alinhac.