The Cauchy problem for the compressible Euler equation is discussed with compactly supported initials. To establish the localexistence of classical solutions by the aid of the theory of quasilinear symmetric hyperbolic systems, a new symmetrization is introduced which works for initials having compact support or vanishing at infinity. It is further shown that as far as the classical solution is concerned, its support does not change, and that the life span is finite for any solution except for the trivial zero solution.
- compactly supported solution
- compressible Euler equation
- non-existence of global solution
- quasi-linear symmetric hyperbolic system
ASJC Scopus subject areas
- Applied Mathematics