Sound-speed variables for polytropic Euler

Assume Countable Choice. Let $d\in \{1,2,3\}$, $\kappa >0$, $\gamma >1$, and put $a=(\gamma -1)\mathrm{/}2$. For positive density $\rho >0$, define the polytropic pressure $p(\rho )=\kappa {\rho }^{\gamma }$, sound speed $c(\rho )=\sqrt{p^{\mathrm{\prime }}(\rho )}=\sqrt{\kappa \gamma }{\rho }^a$, and sound-speed variable $r=c\mathrm{/}a$. The map $\rho \mapsto r$ is a smooth bijection $(0,\mathrm{\infty })\to (0,\mathrm{\infty })$ with inverse $\rho =(ar\mathrm{/}\sqrt{\kappa \gamma }{)}^{1\mathrm{/}a}$. For a velocity $u\in {\mathbb{R}}^d$, set $q=(r,u_1,\dots ,u_d)\in {\mathbb{R}}^{d+1}$. Let $e_0,e_1,\dots ,e_d$ be the standard basis of ${\mathbb{R}}^{d+1}$, with $e_0$ the scalar coordinate and $e_j$ the velocity coordinate $u_j$. Define the real matrix $A_j(q)=u_j{I}_{d+1}+ar(e_0{e}_j^{\mathsf{T}}+e_j{e}_0^{\mathsf{T}}),1\le j\le d\mathrm{.}$ Each $A_j(q)$ is symmetric and linear in $q$; the same formula defines the linear map $A_j:{\mathbb{R}}^{d+1}\to {\mathbb{R}}^{(d+1)\times (d+1)}$ even when the scalar coordinate is nonpositive, as needed for polynomial approximations. To interpret derivatives, let $I\subset \mathbb{R}$ be an open time interval and let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be open, or let $\mathrm{\Omega }=Q_d=(\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}^d$. Take fields $\rho \in C^1(I\times \mathrm{\Omega };(0,\mathrm{\infty }))$ and $u\in C^1(I\times \mathrm{\Omega };{\mathbb{R}}^d)$, and define $r(t,x)=\sqrt{\kappa \gamma }\rho (t,x{)}^a\mathrm{/}a$ and $q(t,x)=(r(t,x),u(t,x))$ pointwise. Spatial derivatives on $Q_d$ are derivatives of the periodic lifts. On a closed time interval the same convention means the fields and first derivatives extend continuously to its endpoints, with one-sided time derivatives there. The first-order system $q_t+{\sum }_{j=1}^d{A}_j(q){\mathrm{\partial }}_{j}q=0$ is a pointwise equality on the time-space domain and means the pair $r_t+u\cdot \mathrm{\nabla }r+ar\mathrm{div}u=0,u_t+u\cdot \mathrm{\nabla }u+ar\mathrm{\nabla }r=0.$ For periodic coefficients, let $q:Q_d\to {\mathbb{R}}^{d+1}$ be componentwise integrable (in particular, any continuous periodic spatial field or time slice above), and take normalized measure $d\mu =(2\pi {)}^{-d}dx$, Fourier coefficients $\widehat{q}(k)=\int q(x)e^{-ik\cdot x}d\mu (x)$, weights $w_k=(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, and $\mathrm{\parallel }q{\mathrm{\parallel }}_{H^m}^2={\sum }_{k\in {\mathbb{Z}}^d}{w}_k^{2m}\mathrm{\mid }\widehat{q}(k){\mathrm{\mid }}^2$, componentwise; real fields satisfy $\widehat{q}(-k)=\overline{\widehat{q}(k)}$. The coefficient integrals are defined on these integrable fields. For general $H^m$ elements, coefficient coordinates mean those of the weighted completion in the Fourier-space definition; no integral formula is imposed on an element without an integrable representative. These are the periodic Fourier-space conventions used for $H^m$ in this development.

Periodic fluid Fourier Sobolev spaces

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

François Golse, The Boltzmann Equation and Its Hydrodynamic Limits, §7.4