Non-existence of global smooth solutions to symmetrizable nonlinear hyperbolic systems

Abstract

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a su± cient condition on the non-existence of global C 1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible°ow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

On Compressible Vortex Sheets

“…Another possibility occurs when we consider initial data containing vacuum. In this case the singularity develops near the interface separating the gas and the vacuum and can appear in the solution before the shock waves, see [34], [50]. New achievements about the free boundary problem for perfect compressible and incompressible fluids in vacuum may be found in [11,32,33].…”