Header

UZH-Logo

Maintenance Infos

Transport equations with integral terms: existence, uniqueness and stability


De Lellis, Camillo; Gwiazda, Piotr; Świerczewska-Gwiazda, Agnieszka (2016). Transport equations with integral terms: existence, uniqueness and stability. Calculus of Variations and Partial Differential Equations, 55(5):online ahead of print.

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. At the end of the note we also point out a relevant example about the strong stability of the continuity equation, which highlights the role of an important assumption in our main stability statement.

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. At the end of the note we also point out a relevant example about the strong stability of the continuity equation, which highlights the role of an important assumption in our main stability statement.

Statistics

Altmetrics

Downloads

7 downloads since deposited on 15 Mar 2017
7 downloads since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:October 2016
Deposited On:15 Mar 2017 15:13
Last Modified:09 Dec 2017 00:24
Publisher:Springer
ISSN:0944-2669
Publisher DOI:https://doi.org/10.1007/s00526-016-1049-9

Download

Download PDF  'Transport equations with integral terms: existence, uniqueness and stability'.
Preview
Content: Accepted Version
Language: English
Filetype: PDF
Size: 355kB
View at publisher