Delayed singularity formation in 2D compressible flow

The initial value problem for the compressible Euler equations in two space dimensions is studied. Of interest is the lifespan of classical solutions with initial data that is a small perturbation from a constant state. The approach taken is to regard the compressible solution as a nonlinear superposition of an underlying incompressible flow and an irrotational compressible flow. This viewpoint yields an improvement for the lifespan over that given by standard existence theory. The estimate for the lifespan is further improved when the initial data possesses certain symmetry. In the case of rotational symmetry, a result of S. Alinhac is reconsidered. The approach is also applied to the study of the incompressible limit. The analysis combines energy and decay estimates based on vector fields related to the natural invariance of the equations.