Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

Autor(es)

Iftimie, D. ^[1] ; Lopes Filho, M. C. ^[2] ; Nussenzveig Lopes, H. J. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Lyon 1, Univ Lyon, CNRS, Inst Camille Jordan, Villeurbanne - France

^[2] Univ Fed Rio de Janeiro, Inst Matemat, BR-21941909 Rio De Janeiro, RJ - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Communications in Partial Differential Equations; v. 45, n. 2, p. 109-145, FEB 1 2020.
• Citações Web of Science: 1

Resumo

In this article we examine the interaction of incompressible 2D flows with material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data, in the case of bounded domains. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulation around each connected component of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Existence of a weak solution for the 2D Euler equations, in velocity form, is guaranteed by Delort's Theorem, when the initial vorticity is a bounded measure satisfying a sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Despite their being equivalent, the qualitative information concerning weak solutions is more apparent from the weak vorticity formulation than from the velocity formulation, and the remainder of the article is devoted to several consequences which can be derived from our main result. First, we consider weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove a set of inequalities which restrict the possible generation of vorticity by the boundary. Next, we prove that, if the weak solution conserves circulation around the boundary components, then it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation around the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. Finally, we discuss the definition of the net mechanical force which the flow exerts on each material boundary component and its relation with conservation of circulation. (AU)

• Processo FAPESP: 07/51490-7 - Aspectos matemáticos da dinâmica dos fluidos incompressíveis
• Beneficiário: Milton da Costa Lopes Filho
• Modalidade de apoio: Auxílio à Pesquisa - Temático