Homogeneous translating polytropic fluid

Assume the Axiom of Countable Choice. Let $d\in \{1,2,3\}$ and $L>0$ measured in metres, and write ${\mathbb{T}}_L^d=(\mathbb{R}\mathrm{/}(2\pi L\mathbb{Z}){)}^d$ for the coordinate torus. When $d=3$ it is the physical cell; when $d=1,2$ it is the reduced coordinate torus and the physical cell is ${\mathbb{T}}_L^3$. In SI units, choose a constant density ${\rho }_{\ast }>0$ in kilograms per cubic metre, a constant velocity $V_{\ast }\in {\mathbb{R}}^d$ in metres per second, a constant ${\kappa }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}>0$ in pascals times $({\mathrm{m}}^3\mathrm{/}\mathrm{k}\mathrm{g}{)}^{\gamma }$, and a dimensionless exponent $\gamma >1$. Define $W_{\ast }\in {\mathbb{R}}^3$ by $W_{\ast }=V_{\ast }$ for $d=3$ and $W_{\ast }=(V_{\ast ,1},\dots ,V_{\ast ,d},0,\dots ,0)$ for $d=1,2$. Identify fields on the three-dimensional torus with their smooth $2\pi L$-periodic lifts to ${\mathbb{R}}^3$ when applying the local continuum equations. In the unforced three-dimensional polytropic Euler model with $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\kappa }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\rho }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\gamma }$ and adopted specific internal energy $e_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\rho )={\kappa }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\rho }^{\gamma -1}\mathrm{/}(\gamma -1)$, take initial data ${\rho }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(0,X)={\rho }_{\ast }$ and $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(0,X)=W_{\ast }$. The constant fields ${\rho }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(t,X)={\rho }_{\ast }$, $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(t,X)=W_{\ast }$, and $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(t,X)={\kappa }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\rho }_{\ast }^{\gamma }$ form a smooth solution for every $t\ge 0$. For $d=1,2$ their reduced representation on ${\mathbb{T}}_L^d$ has velocity $V_{\ast }$. The total energy density $\frac{1}{2}{\rho }_{\ast }\mathrm{\mid }{W}_{\ast }{\mathrm{\mid }}^2+{\rho }_{\ast }{e}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\rho }_{\ast })=\frac{1}{2}{\rho }_{\ast }\mathrm{\mid }{W}_{\ast }{\mathrm{\mid }}^2+{\kappa }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\rho }_{\ast }^{\gamma }\mathrm{/}(\gamma -1)$, measured in pascals, is constant in space and time and equals the reduced expression with $\mathrm{\mid }{V}_{\ast }{\mathrm{\mid }}^2$. For $d=3$ this is the unique local solution branch; for $d=1,2$ it is the unique branch in the transversely invariant reduced model.

Local positive-density polytropic model evolution

David Tong, Fluid Mechanics, Chapter 4: Waves.