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Transport equations with integral terms: existence, uniqueness and stability

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.

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Correspondence to Camillo De Lellis.

Additional information

A. Świerczewska-Gwiazda was supported by the project IdP2011/000661. P. Gwiazda is a coordinator of International Ph.D. Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007–2013 (Ph.D. Programme: Mathematical Methods in Natural Sciences). The research of P. Gwiazda has also has received funding from the National Science Centre, Poland, 2014/13/B/ST1/03094.

Communicated by L. Ambrosio.

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De Lellis, C., Gwiazda, P. & Świerczewska-Gwiazda, A. Transport equations with integral terms: existence, uniqueness and stability. Calc. Var. 55, 128 (2016). https://doi.org/10.1007/s00526-016-1049-9

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Mathematics Subject Classification

  • Primary 35F16
  • Secondary: 37C10
  • 82D60