On the existence of weak solutions for the 2D incompressible Euler equations with in-out flow and source and sink points

TL;DR

This paper proves the existence of weak solutions for the 2D incompressible Euler equations with in-out flow, source, sink, and vortex points, extending previous theories to time-dependent domains and point-source modeling.

Contribution

It extends DiPerna-Lions theory to time-dependent domains and models fluid flow with boundary and point sources by deriving the equations as limits of small holes.

Findings

Existence of weak solutions with in-out flow and point sources.

Extension of DiPerna-Lions theory to time-dependent domains.

Model derivation as limits of small holes.

Abstract

Well-posedness for the two dimensional Euler system with given initial vorticity is known since the works of Judovi\v{c}. In this paper we show existence of solutions in the case where we allowed the fluid to enter in and exit from the boundaries and from some points of the fluid domain. In particular we derive the equations of the model as the limit when we replace the points by some small holes. To do that we extend the DiPerna-Lions theory with non-tangent velocity field on the boundary to the case of time-dependent domain, we extend the existence result for the two dimensional Euler system with in-out flow to time-dependent domain and finally we derive the system that models a fluid which is allowed to enter in and exit from the boundary and some points. The solutions are characterized by the presence of source, sink and vortex points.