Euler equations

The compressible Euler equations for gas dynamics are (mass, momentum, internal energy)
$\frac{\partial \rho }{\partial t} + \nabla \cdot \left( \rho \mathbf{u} \right) = 0 \, ,$
$\frac{\partial (\rho u)}{\partial t} + \frac{\partial}{\partial x} \left( \rho u^2 + p \right) + \frac{\partial}{\partial y} \left( \rho u v \right) =0 \, ,$
$\frac{\partial (\rho v)}{\partial t} + \frac{\partial}{\partial x} \left( \rho u v \right) + \frac{\partial}{\partial y} \left( \rho v^2 + p \right) =0 \, ,$
$\frac{\partial E }{\partial t} + \frac{\partial}{\partial x} \left( u(E+p) \right) + \frac{\partial}{\partial y} \left( v(E+p) \right) =0 \, .$
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.