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Diffusive limit for a Boltzmann-like equation with non-conserved momentum (1904.13253v1)
Published 29 Apr 2019 in math-ph, cond-mat.stat-mech, and math.MP
Abstract: 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 $M_{\rho, 0, T}=\frac{\rho}{(2\pi T){3/2}}\exp[{-\frac{|v|2}{2T}}]$, where $\rho$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L2_{x,v}$.