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June  2016, 11(2): 301-311. doi: 10.3934/nhm.2016.11.301

## 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  April 2015 Revised  August 2015 Published  March 2016

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.
Citation: Maria Colombo, Gianluca Crippa, Stefano Spirito. Logarithmic estimates for continuity equations. Networks and Heterogeneous Media, 2016, 11 (2) : 301-311. doi: 10.3934/nhm.2016.11.301
##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2. [2] L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1191-1244. doi: 10.1017/S0308210513000085. [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005. [4] L. Ambrosio, M. Lecumberry and S. Maniglia, Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow, Rend. Sem. Mat. Univ. Padova, 114 (2005), 29-50. [5] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282. doi: 10.1142/S0219891613500100. [6] A. Clop, R. Jiang, J. Mateu and J. Orobitg, Linear transport equations for vector fields with subexponentially integrable divergence, Calc. Var. Partial Differential Equations, 55 (2016), p21, arXiv:1502.05303 doi: 10.1007/s00526-016-0956-0. [7] M. Colombo, G. Crippa and S. Spirito, Renormalized solutions to the continuity equation with an integrable damping term, Calc. Var. PDE, 54 (2015), 1831-1845. doi: 10.1007/s00526-015-0845-y. [8] G. Crippa and C. De Lellis, Estimates for transport equations and regularity of the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. [9] B. Desjardins, A few remarks on ordinary differential equations, Comm. Partial Diff. Eq., 21 (1996), 1667-1703. doi: 10.1080/03605309608821242. [10] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. [11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. [12] P. B. Mucha, Transport equation: Extension of classical results for div $b \in BMO$, J. Differential Equations, 249 (2010), 1871-1883. doi: 10.1016/j.jde.2010.07.015.

show all references

##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math., 158 (2004), 227-260. doi: 10.1007/s00222-004-0367-2. [2] L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1191-1244. doi: 10.1017/S0308210513000085. [3] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, $2^{nd}$ edition, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005. [4] L. Ambrosio, M. Lecumberry and S. Maniglia, Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow, Rend. Sem. Mat. Univ. Padova, 114 (2005), 29-50. [5] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ., 10 (2013), 235-282. doi: 10.1142/S0219891613500100. [6] A. Clop, R. Jiang, J. Mateu and J. Orobitg, Linear transport equations for vector fields with subexponentially integrable divergence, Calc. Var. Partial Differential Equations, 55 (2016), p21, arXiv:1502.05303 doi: 10.1007/s00526-016-0956-0. [7] M. Colombo, G. Crippa and S. Spirito, Renormalized solutions to the continuity equation with an integrable damping term, Calc. Var. PDE, 54 (2015), 1831-1845. doi: 10.1007/s00526-015-0845-y. [8] G. Crippa and C. De Lellis, Estimates for transport equations and regularity of the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. [9] B. Desjardins, A few remarks on ordinary differential equations, Comm. Partial Diff. Eq., 21 (1996), 1667-1703. doi: 10.1080/03605309608821242. [10] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835. [11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. [12] P. B. Mucha, Transport equation: Extension of classical results for div $b \in BMO$, J. Differential Equations, 249 (2010), 1871-1883. doi: 10.1016/j.jde.2010.07.015.
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