Smooth positive-density barotropic Euler model

Let $I\subset \mathbb{R}$ be an open time interval, let $\mathrm{\Omega }\subset I\times {\mathbb{R}}^3$ be open, and let $P:(0,\mathrm{\infty })\to \mathbb{R}$ be a continuously differentiable pressure function. A smooth positive-density barotropic Euler solution is a pair of fields $\rho \in C^2(\mathrm{\Omega },(0,\mathrm{\infty }))$ and $u\in C^2(\mathrm{\Omega },{\mathbb{R}}^3)$ satisfying ${\mathrm{\partial }}_t\rho +\mathrm{div}(\rho u)=0,\rho ({\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u)+\mathrm{\nabla }P(\rho )=0.$ Here $\mathrm{div}u={\sum }_{i=1}^3{\mathrm{\partial }}_{x_i}{u}_i$, $\mathrm{\nabla }$ is the spatial gradient, and $D_{t}u={\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u$ is the material acceleration. The constitutive stress is $\sigma =-P(\rho )I_3$ under the tension-positive traction convention. Thus “barotropic” means that pressure is a specified function of density alone, and “Euler” here means that the viscous stress is absent. No sign of $P^{\mathrm{\prime }}(\rho )$ is built into the definition; a positive acoustic response is a separate hypothesis where used. In SI interpretation, $\rho$ is in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $u$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $P$ in $\mathrm{P}\mathrm{a}$, and $P^{\mathrm{\prime }}(\rho )$ in ${\mathrm{m}}^2{\mathrm{s}}^{-2}$. The model gives equations for a smooth solution and asserts no existence theorem.

David Tong, Fluid Mechanics, Part I lecture notes, §2.1, printed pp. 14–18

An Introduction to Continuum Mechanics, Second Edition, Chapter 6, §6.5.2, printed pp. 243–244