Properties of the Euler equations

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 $v$, $v-s$, $v+s$, 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 $v-s$ and $v+s$ are the highest possible signal speed within a flow with velocity $v$. 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.

Figure 1: The domain of influence in the $(x,y)$ space

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 $v-s$ and $v+s$, or in other words $v-s$ and $v+s$ are the shock speeds.