Logarithmic estimates for continuity equations

Maria Colombo; Gianluca  Crippa; Stefano  Spirito; Maria Colombo; Gianluca  Crippa; Stefano  Spirito

doi:10.3934/nhm.2016.11.301

Networks and Heterogeneous Media

2016, Volume 11, Issue 2: 301-311. doi: 10.3934/nhm.2016.11.301

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Logarithmic estimates for continuity equations

1.
Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa
2.
Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel
3.
GSSI - Gran Sasso Science Institute, Viale Francesco Crispi 7, 67100 L'Aquila

Received: 01 April 2015 Revised: 01 August 2015
Primary: 35F16; Secondary: 37C10.

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6,9,12], where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.
- Continuity and transport equations,
- well-posedeness,
- Lagrangian flows,
- renormalized solutions
Citation: Maria Colombo, Gianluca Crippa, Stefano Spirito. Logarithmic estimates for continuity equations[J]. Networks and Heterogeneous Media, 2016, 11(2): 301-311. doi: 10.3934/nhm.2016.11.301

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Abstract

The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6,9,12], where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions.

References

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