## Characteristic decomposition of the $$2\times 2$$ quasilinear strictly hyperbolic systems.(English)Zbl 1246.35126

Summary: This paper is devoted to extending the well-known result on reducible equations in R. Courant and K. O. Friedrichs’ book [Supersonic flow and shock waves. Pure Appl. Math. I. New York: Interscience Publ. (1948; Zbl 0041.11302)], that any hyperbolic state adjacent to a constant state must be a simple wave. The authors establish a nice sufficient condition for the existence of characteristic decompositions to the general $$2\times 2$$ quasilinear strictly hyperbolic systems. These decompositions allow for a proof that any wave adjacent to a constant state is a simple wave, despite the fact that the coefficients depend on the independent variables. Consequently, as applications, the authors obtain the same results for the pseudo-steady Euler equations.

### MSC:

 35L60 First-order nonlinear hyperbolic equations 35Q31 Euler equations