Flows of vector fields with point singularities and the vortex-wave system
Gianluca Crippa - Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland (email)
Abstract: 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 $L^p$ 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.
Keywords: Euler equations, vortex-wave system, regular Lagrangian flows, continuity and transport equations, renormalized solutions, singular vector fields.
Received: April 2015; Revised: August 2015; Available Online: October 2015.
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