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Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks. (English) Zbl 1496.35288

Summary: A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to V. I. Yudovich’s paper [“A two-dimensional non-stationary problem on the flow of an ideal incompressible fluid through a given region”, Mat. Sb. 64(106), 562–588 (1964)]. 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 \([1,+\infty]\), 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 \(p > \frac{4}{3}\), in the renormalized sense in the case where \(p > 1\), and in a symmetrized sense in the case where \(p =1\).

MSC:

35Q31 Euler equations
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76F25 Turbulent transport, mixing
35D30 Weak solutions to PDEs
35B45 A priori estimates in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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