Well-posedness of a system of transport and diffusion equations in space of measures

Abstract

In this paper, we establish the well-posedness of the following system of two transport equations coupled with a diffusion equation: ∂tμt1+∇⋅(v1[μt]μt1)=N1(t,μt),∂tμt2−Δμt2=N2(t,μt),∂tμt3+∇⋅(v2[μt]μt3)=N3(t,μt), in Rd where μt1,μt2,μt3 are finite signed measures. Here, the vector field v1, v2 and the source term N1,N2,N3 depend on the measure-valued solution vector μt=(μt1,μt2,μt3).