Euler equations under weighted local-equilibrium convergence

Let $T_{\ast }>0$ seconds, let $L_1,L_2,L_3>0$ be fixed spatial periods in metres, and set $\mathrm{\Omega }={\prod }_{j=1}^3(\mathbb{R}\mathrm{/}{L}_j\mathbb{Z})$. Give $\mathrm{\Omega }$ the completed unnormalised pushforward $dx$ of Lebesgue measure on $C={\prod }_j[0,L_j)$ under the quotient map, as in the kinetic-density definition. Let $v_{\ast }=1\mathrm{m}{\mathrm{s}}^{-1}$. For $k=2,3$ put ${\omega }_k(v):={\sum }_{j=0}^k(\mathrm{\mid }v\mathrm{\mid }\mathrm{/}{v}_{\ast }{)}^j\mathrm{.}$ Let $Q_{\mathrm{e}\mathrm{l}}=\{(v,w,v^{\mathrm{\prime }},w^{\mathrm{\prime }})\in {\mathbb{R}}^{12}:v+w=v^{\mathrm{\prime }}+w^{\mathrm{\prime }},\mathrm{\mid }v{\mathrm{\mid }}^2+\mathrm{\mid }w{\mathrm{\mid }}^2=\mathrm{\mid }{v}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2\}$ be the elastic collision set, and use the nonnegative countably additive symmetric collision measure $\nu$ and operator $Q_{\nu }:{\mathcal{D}}_{\nu }\to L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ specified in the earlier elastic-collision definition; the three particle-exchange and incoming/outgoing invariances are hypotheses on $\nu$. Thus, for each $f^{ϵ}(x,\cdot ,t)$ in its declared collision domain, the weak identity has factor $1\mathrm{/}4$ and collision bracket $\delta \psi =\psi (v)+\psi (w)-\psi (v^{\mathrm{\prime }})-\psi (w^{\mathrm{\prime }})$ whenever the test and both integrals are admissible. For each dimensionless $ϵ>0$, let $f^{ϵ}>0$ be periodic, $C^1$ in $(t,x)$ and $C^2$ in $v$ on the closed slab (with one-sided time derivatives at its endpoints), with $\int {\omega }_3{f}^{ϵ}dv<\mathrm{\infty }$ at each $(t,x)$, and solve on $0<t<T_{\ast }$ $ϵ({\mathrm{\partial }}_t{f}^{ϵ}+v\cdot {\mathrm{\nabla }}_x{f}^{ϵ})=Q_{\nu }(f^{ϵ})\mathrm{.}$ Assume the common $L^1(dv)$ derivative envelopes from the moment-balance proposition: for each multiplier $\psi \in \{m,m{v}_1,m{v}_2,m{v}_3,m\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}2\}$, the time derivative and each spatially weighted derivative have one velocity majorant independent of $(t,x)$ on compact time slabs. At every $(t,x)$, require $f^{ϵ}(x,\cdot ,t)\in {\mathcal{D}}_{\nu }$ and $1,v_1,v_2,v_3,\mathrm{\mid }v{\mathrm{\mid }}^2\in {\mathcal{S}}_{f^{ϵ}(x,\cdot ,t)}$ with the measurability and absolute-convergence conditions of the elastic-collision definition; assume finite velocity moments through degree three. The elastic bracket of each such invariant is zero. Let $n,u,T$ be $C^1$ on $[0,T_{\ast }]\times \mathrm{\Omega }$, periodic in $x$, with $n,T$ bounded above and bounded away from zero and $u$ bounded; $n$ is number density in ${\mathrm{m}}^{-3}$, $u$ is velocity in $\mathrm{m}{\mathrm{s}}^{-1}$, and $T$ is temperature in kelvins. Use the fixed particle mass $m>0$ in kilograms and Boltzmann constant $k_B>0$ in joules per kelvin from the kinetic-density definition, and define $\rho =mn$, $p=n{k}_{B}T$, $E=\rho \mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2+3n{k}_{B}T\mathrm{/}2$, and $M_{n,u,T}(v):=n(2\pi \theta {)}^{-3\mathrm{/}2}{e}^{-\mathrm{\mid }v-u{\mathrm{\mid }}^2\mathrm{/}(2\theta )},\theta :=k_{B}T\mathrm{/}m\mathrm{.}$ Assume, as $ϵ\downarrow 0$, that ${\int }_0^{T_{\ast }}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^3}{\omega }_3(v)\mathrm{\mid }{f}^{ϵ}-M_{n,u,T}\mathrm{\mid }dvdxdt⟶0,$ and ${\int }_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^3}{\omega }_2(v)\mathrm{\mid }{f}^{ϵ}(0,x,v)-M_{n,u,T}(0,x,v)\mathrm{\mid }dvdx⟶0.$ Then $\rho =mn$, $p=n{k}_{B}T$, and $E=\rho \mathrm{\mid }u{\mathrm{\mid }}^2\mathrm{/}2+3n{k}_{B}T\mathrm{/}2$ satisfy the compressible monatomic Euler equations in distributions on $(0,T_{\ast })\times \mathrm{\Omega }$; explicitly, for all scalar test functions $\varphi ,\eta \in C^1([0,T_{\ast }]\times \mathrm{\Omega })$ and vector test function $\zeta \in C^1([0,T_{\ast }]\times \mathrm{\Omega };{\mathbb{R}}^3)$ that are periodic in $x$ and vanish near $t=T_{\ast }$ but may be nonzero at $t=0$, ${\mathrm{\partial }}_t\rho +{\mathrm{\nabla }}_x\cdot (\rho u)=0,$ ${\mathrm{\partial }}_t(\rho u)+{\mathrm{\nabla }}_x\cdot (\rho u\otimes u+p{I}_3)=0,$ ${\mathrm{\partial }}_{t}E+{\mathrm{\nabla }}_x\cdot ((E+p)u)=0,$ and the corresponding weak identities with the initial traces are ${\int }_0^{T_{\ast }}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho {\mathrm{\partial }}_t\varphi +\rho u\cdot {\mathrm{\nabla }}_x\varphi )dxdt+{\int }_{\mathrm{\Omega }}\rho (0,x)\varphi (0,x)dx=0,$ ${\int }_0^{T_{\ast }}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(\rho u\cdot {\mathrm{\partial }}_t\zeta +(\rho u\otimes u+p{I}_3):{\mathrm{\nabla }}_x\zeta )dxdt+{\int }_{\mathrm{\Omega }}\rho (0,x)u(0,x)\cdot \zeta (0,x)dx=0,$ ${\int }_0^{T_{\ast }}\text{\hspace{0.17em}⁣}{\int }_{\mathrm{\Omega }}(E{\mathrm{\partial }}_t\eta +(E+p)u\cdot {\mathrm{\nabla }}_x\eta )dxdt+{\int }_{\mathrm{\Omega }}E(0,x)\eta (0,x)dx=0,$ where $(A:{\mathrm{\nabla }}_x\zeta ):={\sum }_{i,j}{A}_{ij}{\mathrm{\partial }}_{x_j}{\zeta }_i$. Their initial traces are the values of these fields at $t=0$. Since the limiting fields are $C^1$, the equations hold pointwise for $0<t<T_{\ast }$. This is a conditional passage theorem: it assumes the weighted convergence and asserts no such convergence or kinetic solution exists.

Golse, §§3.1–3.2, collision and moment context, full chapter

Zamponi, §3.2, full lecture notes