Transport equation with integral terms

TL;DR

This paper establishes fundamental mathematical properties of solutions to a class of transport equations with integral terms, crucial for modeling complex polymeric flows and related fluid dynamics problems.

Contribution

It introduces a novel Lagrangian framework to analyze transport equations with integral dependencies, extending beyond traditional approaches.

Findings

Proved existence and uniqueness of solutions.

Established stability and compactness properties.

Lays groundwork for future analysis of polymeric flow models.

Abstract

We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate 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.