On characterization and stability of steady state solutions to the hydrostatic Euler equations

Abstract

In many applications in oceanography, meteorology, geophysical flows, blood circulation of the human body or pipeline transport, one may need to study the flow that moves in a narrow domain, in which the horizontal length scale is much larger than other scales. One of the standard approaches to study these problems is called the hydrostatic approximation. For example, one may study the incompressible Euler equations in a very narrow two-dimensional domain with the slip boundary condition. In such a situation, one can alternatively study the hydrostatic Euler equations, which formally describe the leading order behaviour of an incompressible inviscid flow, moving in a narrow channel. In this thesis, we will first recall the background and the formal derivation of the hydrostatic Euler equations in Chapter 1 and 2. Then, in Chapter 3, we will discuss Hamel and Nadirashvili’s recent results (Communications on Pure and Applied Mathematics 70(3):590-608, 2017) for the characterisation of steady state solutions to the incompressible Euler equations. After that, some details of Xie and Xin’s proof (SIAM Journal on Mathematical Analysis, 42(2):751-784, 2010) of the existence of global steady subsonic Euler flows through infinitely long nozzles will be presented. Regarding the steady hydrostatic Euler equations, new results of characterisation and existence will be shown. More precisely, we will show how to apply Hamel and Nadirashvili’s method to prove a similar characterisation of steady state solutions to the hydrostatic Euler equations in Chapter 4 and adapt Xie and Xin’s approach to prove the existence of global steady solutions to the hydrostatic Euler equations in perturbed domains in Chapter 5 and 6.