Euler equations

The compressible Euler equations for gas dynamics are (mass, momentum, internal energy)
${\frac {\partial \rho }{\partial t}}+\nabla \cdot \left(\rho \mathbf {u} \right)=0\,,$
${\frac {\partial (\rho u)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho u^{2}+p\right)+{\frac {\partial }{\partial y}}\left(\rho uv\right)=0\,,$
${\frac {\partial (\rho v)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\rho uv\right)+{\frac {\partial }{\partial y}}\left(\rho v^{2}+p\right)=0\,,$
${\frac {\partial E}{\partial t}}+{\frac {\partial }{\partial x}}\left(u(E+p)\right)+{\frac {\partial }{\partial y}}\left(v(E+p)\right)=0\,.$
A suitable equation of state used to close the system is given by
$E={\frac {p}{\gamma -1}}+{\frac {\rho }{2}}\left(u^{2}+v^{2}\right)\;,$
where typically $\gamma =1.4$ for a monoatomic gas.