Logarithmic estimates for continuity equations

Maria Colombo; Gianluca  Crippa; Stefano  Spirito

doi:10.3934/nhm.2016.11.301

Article Contents

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

This issue Previous Article A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation Next Article A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case

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: March 31, 2015

Revised: July 31, 2015

Published: May 2016

Abstract Related Papers Cited by

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.

Keywords:

Mathematics Subject Classification: Primary: 35F16; Secondary: 37C10.

Citation:

Related Papers

Cited by

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:1502.05303doi: 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.