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

Qingshan Chen

doi:10.3934/dcds.2019133

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2019, Volume 39, Issue 6: 3215-3237. Doi: 10.3934/dcds.2019133

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

Qingshan Chen^,

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

* Corresponding author: Qingshan Chen
* Corresponding author: Qingshan Chen

Received: May 31, 2018

Revised: October 31, 2018

Published: February 2019

The first author is in part supported by Simons Foundation (319070)

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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.

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Mathematics Subject Classification: Primary: 35D30, 76B03; Secondary: 86A05, 86A10.

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