Flows of vector fields with point singularities and the vortex-wave system

Crippa, Gianluca and Lopes Filho, Milton and Miot, Evelyne and Nussenzveig Lopes, Helena. (2016) Flows of vector fields with point singularities and the vortex-wave system. Discrete and continuous dynamical systems, 36 (5). pp. 2405-2417.

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Official URL: http://edoc.unibas.ch/40180/

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The vortex-wave system is a version of the vorticity equation governing the motion of 2D incompressible fluids in which vorticity is split into a finite sum of Diracs, evolved through an ODE, plus an Lp part, evolved through an active scalar transport equation. Existence of a weak solution for this system was recently proved by Lopes Filho, Miot and Nussenzveig Lopes, for p>2, but their result left open the existence and basic properties of the underlying Lagrangian flow. In this article we study existence, uniqueness and the qualitative properties of the (Lagrangian flow for the) linear transport problem associated to the vortex-wave system. To this end, we study the flow associated to a two-dimensional vector field which is singular at a moving point. We first observe that existence and uniqueness of the regular Lagrangian flow are ensured by combining previous results by Ambrosio and by Lacave and Miot. In addition we prove that, generically, the Lagrangian trajectories do not collide with the point singularity. In the second part we present an approximation scheme for the flow, with explicit error estimates obtained by adapting results by Crippa and De Lellis for Sobolev vector fields.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)
UniBasel Contributors:Crippa, Gianluca
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:American Institute of Mathematical Sciences
Note:Publication type according to Uni Basel Research Database: Journal article
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edoc DOI:
Last Modified:11 Oct 2017 12:09
Deposited On:05 Apr 2016 12:10

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