Inserted: 8 jun 2016
Last Updated: 24 may 2017
Journal: Calc. Var. Partial Differential Equations
We prove some theorems on the existence, uniqueness, stability and com- pactness properties of solutions to inhomogeneous transport equations with Sobolev co- efficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to for- mulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will provide, in a separate paper, a sequential stability theorem for the full problem of the flow of concentrated polymers.