We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_t u + div(bu) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non- autonomous vector fields $b$ with bounded divergence.

A uniqueness result for the continuity equation in two dimensions

ALBERTI, GIOVANNI;
2014

Abstract

We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_t u + div(bu) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$. As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non- autonomous vector fields $b$ with bounded divergence.
Alberti, Giovanni; Stefano, Bianchini; Gianluca, Crippa
File in questo prodotto:
File Dimensione Formato  
abc-uniqueness-v19.2.pdf

accesso aperto

Tipologia: Documento in Post-print
Licenza: Creative commons
Dimensione 439.94 kB
Formato Adobe PDF
439.94 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11568/245396
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 39
  • ???jsp.display-item.citation.isi??? 35
social impact