We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure rho(1, b), where rho is an element of M+(Rd+1) and b : Rd+1 -> R-d is a rho-integrable vector field with div(t,x)(rho(1, b)) = mu is an element of M(R x R-d): forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE div(t,x)(rho(1, b)) = mu on a partition of R+ x R-d obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation.

FORWARD UNTANGLING AND APPLICATIONS TO THE UNIQUENESS PROBLEM FOR THE CONTINUITY EQUATION / Bianchini, S; Bonicatto, P. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 41:6(2021), pp. 2739-2776. [10.3934/dcds.2020384]

FORWARD UNTANGLING AND APPLICATIONS TO THE UNIQUENESS PROBLEM FOR THE CONTINUITY EQUATION

Bianchini, S;Bonicatto, P
2021

Abstract

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure rho(1, b), where rho is an element of M+(Rd+1) and b : Rd+1 -> R-d is a rho-integrable vector field with div(t,x)(rho(1, b)) = mu is an element of M(R x R-d): forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE div(t,x)(rho(1, b)) = mu on a partition of R+ x R-d obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation.
41
6
2739
2776
Bianchini, S; Bonicatto, P
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/128616
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