Differentiability in Measure of the Flow Associated with a Nearly Incompressible BV Vector Field

Bianchini S, De Nitti N (2022)

Publication Type: Journal article

Publication year: 2022

Journal

Archive for Rational Mechanics and Analysis Springer Verlag (Germany)

DOI: 10.1007/s00205-022-01820-1

Abstract

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∞(Rd+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.

Authors with CRIS profile

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

Involved external institutions

Scuola Internazionale Superiore di Studi Avanzati (SISSA) IT Italy (IT)

How to cite

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://dx.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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