Ideal-gas compressible Navier–Stokes–Fourier model

Fix an integer $d\ge 1$, a gas constant $R>0$, an adiabatic exponent $\gamma >1$, a constant reference entropy $s_0\in \mathbb{R}$, shear viscosity $\mu >0$, bulk viscosity $\zeta \ge 0$, and heat conductivity $\kappa >0$. Put $c_v=R\mathrm{/}(\gamma -1)$. A conservative state is $U=(\rho ,m,E)\in \mathbb{R}\times {\mathbb{R}}^d\times \mathbb{R}$. Its admissible state set is

$\mathcal{G}=\{(\rho ,m,E):\rho >0,E>\frac{\mathrm{\mid }m{\mathrm{\mid }}^2}{2\rho }\}\mathrm{.}$

For $U\in \mathcal{G}$, define the velocity $u=m\mathrm{/}\rho$, internal-energy density $\epsilon =E-\mathrm{\mid }m{\mathrm{\mid }}^2\mathrm{/}(2\rho )$, pressure $p=(\gamma -1)\epsilon$, temperature $T=p\mathrm{/}(R\rho )$, specific entropy $s(U)=c_v\log (p\mathrm{/}{\rho }^{\gamma })+s_0$, and mathematical entropy $\eta (U)=-\rho s(U)$. The Euler flux in coordinate direction $j$ is

$F_j(U)=(\rho u_j,m{u}_j+p{e}_j,(E+p)u_j),j=1,\dots ,d,$

where $e_j$ is the $j$th standard basis vector of ${\mathbb{R}}^d$. The mathematical entropy flux is $q_j(U)=\eta (U)u_j$.

For a differentiable velocity field, write $D=\mathrm{sym}\mathrm{\nabla }u$ with $D_{ab}=({\mathrm{\partial }}_b{u}_a+{\mathrm{\partial }}_a{u}_b)\mathrm{/}2$, and $D^0=D-(\mathrm{tr}D\mathrm{/}d)I$. The Newtonian stress and Fourier viscous flux are

$\tau =2\mu D^0+\zeta (\mathrm{tr}D)I,F_j^{\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}}=(0,{\tau }_{j1},\dots ,{\tau }_{jd},u\cdot {\tau }_{j\cdot }+\kappa {\mathrm{\partial }}_{j}T)\mathrm{.}$

The compressible Navier–Stokes–Fourier balance law is the conservative system

${\mathrm{\partial }}_{t}U+{\sum }_{j=1}^d{\mathrm{\partial }}_j{F}_j(U)={\sum }_{j=1}^d{\mathrm{\partial }}_j{F}_j^{\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{c}}(U,\mathrm{\nabla }u,\mathrm{\nabla }T)\mathrm{.}$

Unless a boundary is specified separately, a periodic solution means that all fields and fluxes are periodic in each spatial coordinate. The positive parameters and the inequalities defining $\mathcal{G}$ imply $p>0$ and $T>0$.