1.1.3 The solution of the Euler equations

The task is now: Find a solution for Eq. (2) for given initial conditions

$$\left( \rho, \rho\ensuremath{\mathchoice{\mbox{\boldmath$\d... ...ox{\boldmath$\scriptscriptstyle x$}}}, t_0 \right) \enspace ,$$ (3)

boundary conditions

$$\left( \rho, \rho\ensuremath{\mathchoice{\mbox{\boldmath$\d... ...ptscriptstyle x$}}}_\mathrm{boundaries}, t \right) \enspace ,$$ (4)

and material function (equation of state)

$$P= P\left( \rho, e_{\rm i}\right)$$ (5)

with

$$\textstyle e_{\rm i}= e_{\rm ik}- \frac{1}{2} \ensuremath{... ...yle v$}} {\mbox{\boldmath$\scriptscriptstyle v$}}} \enspace .$$ (6)