Estimates of the transition density of a gas system

AbstractLet X be the diffusion Markov process on Rd with the generator L=12∑i,j=1daij(x)∂xixj2+∑i=1dbi(x)∂xi, and transition density G(t,x,y). Under some conditions on the matrix a(x) we get the estimate sup0<t<T,x,y∈Rdt|∇xG(t,x,y)|G(2t,x,y)<+∞ for all T>0. The latter estimate is used to get the existence and uniqueness of a solution of the following gas system: S(a,b)∂t(ρ)+div(uρ)=L∗(ρ),∂t(uiρ)+div(uiuρ)=L∗(uiρ),∀1⩽i⩽d,ρ(dx,t)→ρ0(dx),ui(x,t)ρ(dx,t)→vi(x)ρ0(dx)weakly, ast→0+, where ρ0(dx) (a probability measure on Rd), and the bounded vector field v:=(v1,…,vd):Rd→Rd are given. The family of probability measures ρ:=ρ(dx,t) and the velocities u:=u(x,t) are unknown. Here L∗ is the formal adjoint operator of L