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Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with L 1 Vorticity

Crippa, Gianluca and Nobili, Camilla and Seis, Christian and Spirito, Stefano. (2017) Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with L 1 Vorticity. SIAM Journal on Mathematical Analysis, 49 (5). pp. 3973-3998.

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Abstract

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an L 1 function, extending the Lagrangian theory in [ 6 ]. The proof is based on a combination of a stability estimate via optimal transport techniques developed in [ 28 ] and some tools from harmonic analysis introduced in [ 6 ]. In the second part of the paper, we address a question that arose in [ 21 ], namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in L 1 , extending the proof for the L p case in [ 11 ].
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
UniBasel Contributors:Crippa, Gianluca and Nobili, Camilla
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Society for Industrial and Applied Mathematics
ISSN:0036-1410
Note:Publication type according to Uni Basel Research Database: Journal article
Language:English
Identification Number:
Last Modified:26 Jan 2018 15:31
Deposited On:26 Jan 2018 15:31

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