Diffusion limit for non homogeneous and non-micro-reversible processes (Q2710414)

The authors investigate the behavior of solutions \(f_{\varepsilon}(t,x,v)\) of linear kinetic transport equations NEWLINE\[NEWLINE\partial_t f_{\varepsilon} + {1\over{\varepsilon}}a(v))\cdot\nabla_x f_{\varepsilon} ={1\over{\varepsilon^2}}L(f_{\varepsilon}) NEWLINE\]NEWLINE where \(a(v)\) and \(L\) are considered in great generality. In particular, no detailed balance principle is assumed for \(L,\) the scattering kernels in \(L\) can be \(x-\)dependent, and so can the equilibria given by \(L(F)=0.\) Under very general assumptions on \(L\) and \(a\) the existence of suitable equilibria is shown as a consequence of the Krein-Rutman theorem, and as \(\varepsilon \searrow 0,\) a subsequence of \(\rho_{\varepsilon}(t,x):=\int f_{\varepsilon} (t,x,v) d \mu (v)\) is shown to converge strongly in \(L^2(0,T;L^2_{loc}({\mathbb R}^N))\) to a solution of the drift-diffusion equation NEWLINE\[NEWLINE\partial_t\rho+ \text{div}_x(A(x)\rho -D(x)\nabla_x\rho)=0.NEWLINE\]NEWLINE \(A\) and \(D\) are given in terms of \(L\) and \(F.\)