Bianchini S, De Nitti N (2022)

**Publication Type:** Journal article

**Publication year:** 2022

**DOI:** 10.1007/s00205-022-01820-1

We study the regularity of the flow X(t, y) , which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution ρ∈ L^{∞}(R^{d}^{+}^{1}) of the continuity equation ∂tρ+div(ρb)=0,with b∈Lt1BVx. We prove that X is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),where (Db) y(d t) is the disintegration of the measure ∫Db(t,·)dt with respect to the partition given by the trajectories X(t, y) and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a BV vector field.

Nicola De Nitti
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

**APA:**

Bianchini, S., & De Nitti, N. (2022). Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field. *Archive for Rational Mechanics and Analysis*. https://doi.org/10.1007/s00205-022-01820-1

**MLA:**

Bianchini, Stefano, and Nicola De Nitti. "Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field." *Archive for Rational Mechanics and Analysis* (2022).

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