Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

Abstract

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich's paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed.In this paper we investigate the existence of weak solutions to this system by relying on a priori bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a L p integrability in space, with p in [1, +∞], and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where p > 4 3 , in the renormalized sense in the case where p > 1, and in a symmetrized sense in the case where p = 1.

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

“…As far as we know the literature of existence of solution for the Euler system ( 4) is quite incomplete due to the fact that we consider time dependent domains. For this reason we extend all the existence results proved in [8] in this new setting.…”

“…The existence of regular solutions for the system (39) was done in [8] Lemma 4, in the case the domain F (t) = F (0) does not depend on time. In our setting the extra difficulty is to deal with the fact that F (t) is time dependent.…”

“…Proof. Existence and uniqueness are shown as in Lemma 4 of [8] and the proof is based on a Schauder fixed point argument. Regarding the estimate (40), it follows formally by multiply the first equation of (39) by G ′ (|ω ν |)ω ν /|ω ν |, integrate in (0, T ) ⊙ F (t) and some integrations by part.…”

“…In the case where the fluid domain is time independent, i.e. Ω \ (S + (t) ∪ S − (t)) = Ω \ (S + (0) ∪ S − (0)), existence of solutions to the system (1) was shown in [8] for initial and entering vorticity in L p for p 1. Uniqueness was shown in [20] in the case of L ∞ initial and entering vorticity.…”

“…In the case where the source and the sink are time-independent point, i.e. x + (t) = x + and x − (t) = x − = x + , the system was derived in [8] and it reads…”