On the failure of the chain rule for the divergence of Sobolev vector fields-Reference-Cited by-同舟云学术

On the failure of the chain rule for the divergence of Sobolev vector fields

Published:2023-06 Issue:02 Volume:20 Page:349-385
ISSN:0219-8916
Container-title:Journal of Hyperbolic Differential Equations
language:en
Short-container-title:J. Hyper. Differential Equations

Author:

Buck Miriam¹,Modena Stefano²

Affiliation:

1. Department of Mathematics, Technische Universität Darmstadt, Schloßgartenstraße 7, 64285 Darmstadt, Germany

2. Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 L’Aquila, Italy

Abstract

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension [Formula: see text], for which the chain rule problem has a negative answer. In particular, for any renormalization map [Formula: see text] (satisfying suitable assumptions) and any (distributional) renormalization defect [Formula: see text] of the form [Formula: see text], where [Formula: see text] is an [Formula: see text] vector field, we can construct an incompressible Sobolev vector field [Formula: see text] and a density [Formula: see text] for which [Formula: see text] but [Formula: see text], provided [Formula: see text].

Publisher

World Scientific Pub Co Pte Ltd

Subject

General Mathematics,Analysis

Link

https://www.worldscientific.com/doi/pdf/10.1142/S0219891623500108

Reference23 articles.

1. On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems

2. A uniqueness result for the decomposition of vector fields in $$\mathbb {R}^{{d}}$$

3. Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

4. Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

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