Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - G. Crippa

Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields

created by ambrosio on 17 Jan 2006
modified by crippa on 05 Mar 2009


Published Paper

Inserted: 17 jan 2006
Last Updated: 5 mar 2009

Journal: In: Transport Equations and Multi-D Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana
Volume: 5
Year: 2008


In these notes we illustrate some recent developments of the DiPerna-Lions theory in the case of vector fields having a BV regularity with respect to the spatial variables. In the Sobolev case we discuss also the (weak) differentiability properties of the flow X(t,x) with respect to the x variable, and some recent quantitative regularity estimates leading to a new proof of the existence and the uniqueness of the flow, as well as to quantitative stability results.

Keywords: Flow theory, Approximate differentiability, Maximal functions


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