UC San Diego

The Euler equations are a fundamental yet celebrated set of mathematical equations that describe the motions of inviscid, incompressible fluid on planar domains. They play a critical role in various fields of study including fluid dynamics, aerodynamics, hydrodynamics and so on. Though it was first written out in 1755, there are still many open questions regarding to this rich system, including some fundamental questions of Euler equations on singular domains. Unlike existence of weak solutions that were proven on considerably general domains, uniqueness of such solutions are still quite open on singular domains, even on convex domains. In this thesis, we will show uniqueness of weak solutions on singular domains given two different assumptions of initial vorticity ω0 ∈ L∞:1. ω0 is constant near the boundary. 2. ω0 is constant near the boundary and has a sign (non-positive or non-negative). Under the first assumption, the previous best uniqueness results can only be applied to C1,1 domains except at finitely many corners with interior angles less than π. Here, we will extend the result to fairly general singular domains which are only slightly more restrictive than the exclusion of corners with angles larger than π, thus including all convex domains. We derive this by showing that the Euler particle trajectories cannot reach the boundary in finite time and hence the vorticity cannot be created by the boundary. We will also show that if the given geometric condition is not satisfied, then we can construct a domain and a bounded initial vorticity such that some particle could reach the boundary in finite time. Under the second assumption with the sign condition, the previous best uniqueness result can only be applied to C1,1 domains with finitely many corners with interior angels larger than π/2 . Here, we will extend the result to a class of general singular open bounded simply connected domains, which can be possibly nowhere C1 and there are no restrictions on the size of each angle.