We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the \emph{chain rule problem} has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = \div h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $\rho \in L^p$ for which $\div (\rho u) =0$ but $\div (\beta(\rho) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$.

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

Modena, Stefano
2023-01-01

Abstract

We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the \emph{chain rule problem} has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable assumptions) and any (distributional) renormalization defect $T$ of the form $T = \div h$, where $h$ is an $L^1$ vector field, we can construct an incompressible Sobolev vector field $u \in W^{1, \tilde p}$ and a density $\rho \in L^p$ for which $\div (\rho u) =0$ but $\div (\beta(\rho) u) = T$, provided $1/p + 1/\tilde p \geq 1 + 1/(d-1)$.
2023
Convex integration, chain rule, DiPerna Lions
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/34165
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