Euler equations

Weak solutions of the 2D Euler equations

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B1 L3 R3119

In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.

A Variational Theory of Aero-hydrodynamics

Haithem Taha, Associate Professor

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B5 L5 R5209

The Euler equation does not possess a unique solution for the flow over a two-dimensional object. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a two-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such closure in the literature, even in cases where it is not applicable (e.g. unsteady).