Given bounded vector field $bcolon RR^d o RR^d$, scalar field $ucolon RR^d o RR$ and a smooth function $eta colon RR o RR$ we study the characterization of the distribution $div(eta(u)b)$ in terms of $div b$ and $div(u b)$. In the case of $BV$ vector fields $b$ (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal'y, up to an error term which is a measure concentrated on so-called emph{tangential set} of $b$. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible $BV$ vector field $b$ and a bounded function $u$ for which this term is nonzero. For steady nearly incompressible vector fields $b$ (and under some further assumptions) in case when $d=2$ we provide complete characterization of $div(eta(u) b)$ in terms of $div b$ and $div(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $RR^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $d_t u + b cdot abla u=0$ is emph{renormalized}, i.e. also solves $d_t eta(u) + b cdot abla eta(u)=0$ for any smooth function $eta colonRR o RR$. As a consequence we obtain new uniqueness result for this equation.

Steady nearly incompressible vector fields in 2D: chain rule and renormalization / Bianchini, Stefano; Gusev, N. A.. - In: ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. - ISSN 0003-9527. - 222:2(2016), pp. 451-505. [10.1007/s00205-016-1006-y]

Steady nearly incompressible vector fields in 2D: chain rule and renormalization

Bianchini, Stefano
;
2016-01-01

Abstract

Given bounded vector field $bcolon RR^d o RR^d$, scalar field $ucolon RR^d o RR$ and a smooth function $eta colon RR o RR$ we study the characterization of the distribution $div(eta(u)b)$ in terms of $div b$ and $div(u b)$. In the case of $BV$ vector fields $b$ (and under some further assumptions) such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal'y, up to an error term which is a measure concentrated on so-called emph{tangential set} of $b$. We answer some questions posed in their paper concerning the properties of this term. In particular we construct a nearly incompressible $BV$ vector field $b$ and a bounded function $u$ for which this term is nonzero. For steady nearly incompressible vector fields $b$ (and under some further assumptions) in case when $d=2$ we provide complete characterization of $div(eta(u) b)$ in terms of $div b$ and $div(u b)$. Our approach relies on the structure of level sets of Lipschitz functions on $RR^2$ obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique we obtain new sufficient conditions when any bounded weak solution $u$ of $d_t u + b cdot abla u=0$ is emph{renormalized}, i.e. also solves $d_t eta(u) + b cdot abla eta(u)=0$ for any smooth function $eta colonRR o RR$. As a consequence we obtain new uniqueness result for this equation.
2016
222
2
451
505
https://doi.org/10.1007/s00205-016-1006-y
http://preprints.sissa.it/xmlui/handle/1963/7412
https://link.springer.com/article/10.1007/s00205-016-1006-y
Bianchini, Stefano; Gusev, N. A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/17214
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