Continuity equations and ODE flows with non-smooth velocity
2014 ◽
Vol 144
(6)
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pp. 1191-1244
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Keyword(s):
In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.
1979 ◽
Vol 71
(1)
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pp. 167-186
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1959 ◽
Vol 3
(1)
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pp. 355-380
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1969 ◽
Vol 5
(3)
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pp. 515-530
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2006 ◽
Vol 04
(03)
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pp. 247-262
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1996 ◽
Vol 06
(03)
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pp. 295-314
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1985 ◽
Vol 37
(6)
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pp. 1176-1200
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