Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - S. Nicolussi Golo - F. Serra Cassano

Classical flows of vector fields with exponential or sub-exponential summability

created by nicolussigolo on 03 Aug 2022



Inserted: 3 aug 2022

Year: 2022

ArXiv: 2208.01381 PDF


We show that vector fields $b$ whose spatial derivative $D_xb$ satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if $D_xb$ satisfies a suitable exponential summability condition then the flow associated to $b$ has Sobolev regularity, without assuming boundedness of ${\rm div}_xb$. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations.

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