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 init ial data, we give a complete asymptotic expansion of the lifespan of classi cal solutions, that is, we solve a conjecture posed by John and Hormander. As an application of our result, we show that the solution of three-dimensi onal isentropic compressible Euler equations with irrotational initial data which are a small perturbation from a constant state will develop singular ity in the first-order derivatives in finite time while the solution itself is continuous. Furthermore, for this special case, we also solve a conject ure of Alinhac.