# On the Well-posedness of Reduced $3D$ Primitive Geostrophic Adjustment Model with Weak Dissipation

## TL;DR

This paper establishes well-posedness results for a reduced 3D geostrophic model with weak dissipation, demonstrating global solutions, regularization convergence, and blow-up criteria.

## Contribution

It proves well-posedness for the reduced 3D primitive geostrophic model with weak dissipation and analyzes the regularized model's convergence and blow-up conditions.

## Key findings

- Proved local and global well-posedness for the model.
- Established convergence of Voigt α-regularized solutions.
- Derived criteria for finite-time blow-up.

## Abstract

In this paper we prove the local well-posedness and global well-posedness with small initial data of the strong solution to the reduced $3D$ primitive geostrophic adjustment model with weak dissipation. The term reduced model stems from the fact that the relevant physical quantities depends only on two spatial variables. The additional weak dissipation helps us overcome the ill-posedness of original model. We also prove the global well-posedness of the strong solution to the Voigt $\alpha$-regularization of this model, and establish the convergence of the strong solution of the Voigt $\alpha$-regularized model to the corresponding solution of original model. Furthermore, we derive a criterion for finite-time blow-up of reduced $3D$ primitive geostrophic adjustment model with weak dissipation based on Voigt $\alpha$-regularization.