Global existence to the BGK model of Boltzmann equation (Q909115)

The author has proved results on existence and stability for solutions to the BGK model of Boltzmann equation (1) \[ \partial_ tf+v\cdot \nabla_ xf+f=M[f],\quad t\geq 0,\quad x\in {\mathbb{R}}^ N,\quad v\in {\mathbb{R}}^ N, \] \[ M[f]=(\rho /(2\pi T)^{N/2})\exp (-| v-u|^ 2/(2T)), \] \[ (\rho,\rho u,\rho | u|^ 2+\rho T)(t,x)=\int_{{\mathbb{R}}^ N}(1,v,| v|^ 2)f(t,x,v)dv. \] The proof mainly relies on the strong compactness of \(\rho\), u, T and on a new estimate on the third moment of f: \(\int | v|^ 3f dv\). The entropy relation for (1) is also proved.