The blow-up of solutions for two-dimensional irrotational compressible Euler equations

For two-dimensional irrotational compressible Euler equations with initial data where that is a small perturbation from a constant state, we prove tha t the first-order derivatives of rho. u blow-up at the blow-up time, while rho, v remain continuous. In particular, in the irrotational cause we prove S. Alinhac's statement.