Diffusive limit for a Boltzmann-like equation with non-conserved momentum

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^\varepsilon(x,v,t)=f(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$, as $\varepsilon\to 0$ tends to a Maxwellian $M_{ρ, 0, T}=\fracρ{(2πT)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$, where $ρ$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.

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