In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible threedimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension d ≥ 2 with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic C∞ self-similar solutions with suitable decay at infinity.
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)
- Euler equations
- Self-similar profile