The Barotropic Quasi-Geostrophic Equation under a Free Surface

On the well-posedness of the inviscid multi-layer quasi-geostrophic equations

“…Unlike in the barotropic case (see [8]), the inner product involving the zero order term on the right-hand side does not vanish, thanks to the vertical layer interactions. We proceed by obtaining an estimate on this term.…”

“…For i = 2, 3, the non-standard scalar elliptic BVP with a zero-order term has been considered in [8]. There, with the Green's function for the Helmholtz equation, it is shown that the constant boundary value |l i | on the boundary can be bounded in terms of |q i | ∞ , and the solution ψ i belongs to W 2,p (M) for any p > 1, and is Hölder and quasi-Lipschitz continuous.…”

“…The present work focuses on simpler models, but on more general settings, namely purely inviscid models on arbitrary bounded domains with physically relevant boundary conditions. Th well-posedness of the barotropic QG equation with a free top surface is the subject of a previous work ( [8]). The goal of the present work is to address the issue of well-posedness for the multi-layer QG equations.…”

“…A major challenge in the previous work ( [8]) is the non-homogeneous boundary conditions on the streamfunction, which is imposed to accommodate the free fluctuations of the top surface. There, the challenge is dealt with by the superposition rule and an estimate on the constant non-zero value of the streamfunction.…”

“…It turns out that the interior interfaces behave like the top surface, and can be treated as such. Therefore, the same type of boundary conditions are imposed on the interior interfaces, and they are treated in exactly the same way as in [8].…”