= Solution
Write $v=u_x$ and $D=\partial_t+v\partial_x$. The <Euler equations> and <conservation of mass> give $D\rho=-\rho v_x$ and $\rho D u=-p_x e_x$. The internal-energy <density> of the <ideal gas> is $e=p/(\gamma-1)$. The adiabatic <pressure> equation therefore gives
$$
\partial_te+\partial_x(ve)=-p\partial_xv.
$$
Multiply the momentum equation by $u$ and use continuity to obtain the kinetic-energy balance
$$
\partial_t(\rho u^2/2)+\partial_x(v\rho u^2/2)=-v\partial_xp.
$$
Adding the two equations moves $\partial_x(pv)$ into the flux. Thus the entire system is $\partial_tQ+\partial_xF=0$, where
$$
\boxed{Q=\begin{pmatrix}\rho\\\rho v\\\rho u_y\\\rho u_z\\E\end{pmatrix},\qquad
F=\begin{pmatrix}\rho v\\\rho v^2+p\\\rho v u_y\\\rho v u_z\\v(E+p)\end{pmatrix},\qquad
E=\frac12\rho(v^2+u_y^2+u_z^2)+\frac p{\gamma-1}.}
$$
This derives the <conservative energy flux of a polytropic ideal gas>. In particular one-dimensional spatial dependence does not remove the transverse <velocities> from the total energy. The <energy flux> can equally be written $\rho v(u^2/2+w)$ with <specific enthalpy> $w=\gamma p/[(\gamma-1)\rho]$.
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