The Euler equations are a simpler version of the *Navier-Stokes
equations*. The latter contain terms for the viscosity and thermal
conductivity of the gas. In astrophysics these are normally not
thought to be important.

Using the Euler equations also implies using a *fluid approximation*,
i.e. the particles interact with each other sufficiently to establish
a Maxwell-Boltzmann distribution. This approximation is mostly valid,
but there are exceptions.

From a mathematical point of view the Euler equations are a set of non-linear, coupled, hyperbolic differential equations. Hyperbolic differential equations have two important properties:

- they allow discontinuous solutions; in physical terms this means that the flow can contain shocks or contact discontinuities.
- one can define so called
*characteristics*or*characteristic speeds*. These are the eigenvalues of the problem: the solution can be written in terms of a sum of eigenvectors, three in the case of a one-dimensional problem. The three eigenvectors are also called*waves*and are physically associated with the characteristic speeds , , , the velocity of the flow, and the velocity of sound added and subtracted. The physical relevance of this is that in a gas no signal can travel faster than the local sound speed, and and are the highest possible signal speed within a flow with velocity . This also means that the characteristics delineate a domain of influence in space-time (see Fig. 1). Point Q can only influence the hashed region of space-time.

Physically, shocks can be thought to occur because the particles suddenly have to adjust to a new situation. There is therefore a close relation between the characteristics and the shocks: if for example an explosion occurs at point A, its effect will spread with the characteristic speeds and , or in other words and are the shock speeds.