American Institute of Mathematical Sciences

June  2019, 39(6): 3215-3237. doi: 10.3934/dcds.2019133

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

 Department of Mathematical Sciences, Clemson University, Clemson, SC 29631, USA

* Corresponding author: Qingshan Chen

Received  June 2018 Revised  November 2018 Published  February 2019

Fund Project: The first author is in part supported by Simons Foundation (319070).

The inviscid multi-layer quasi-geostrophic equations are considered over an arbitrary bounded domain. The no-flux but non-homogeneous boundary conditions are imposed to accommodate the free fluctuations of the top and layer interfaces. Using the barotropic and baroclinic modes in the vertical direction, the elliptic system governing the streamfunctions and the potential vorticity is decomposed into a sequence of scalar elliptic boundary value problems, where the regularity theories from the two-dimensional case can be applied. With the initial potential vorticity being essentially bounded, the multi-layer quasi-equations are then shown to be globally well-posed, and the initial and boundary conditions are satisfied in the classical sense.

Citation: Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3215-3237. doi: 10.3934/dcds.2019133
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