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 that the first-order derivatives of , blow-up at the blow-up time, while , remain continuous. In particular, in the irrotational case we prove S. Alinhac's statement.