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A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain

Qingshan Chen

Abstract

The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existence and uniqueness of a generalized solution is proven, provided that the initial potential vorticity (PV) field is essentially bounded. If the initial PV field is differentiable, then the solution is shown to satisfy the system in the classical sense.

A global well-posedness result for the three-dimensional inviscid quasi-geostrophic equation over a cylindrical domain

Abstract

The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on the top and bottom surfaces, while no-flux boundary conditions combined with constant circulations are imposed on the lateral boundary loops. The global existence and uniqueness of a generalized solution is proven, provided that the initial potential vorticity (PV) field is essentially bounded. If the initial PV field is differentiable, then the solution is shown to satisfy the system in the classical sense.
Paper Structure (9 sections, 9 theorems, 119 equations, 1 figure)

This paper contains 9 sections, 9 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Specification of the physical model
  3. Well-posedness and solution regularity of the non-standard three-dimensional elliptic boundary value problem
  4. Existence and uniqueness of a weak solution to the 3D QG
  5. A classical solution
  6. Remarks
  7. Estimates on the Green's function to the non-standard elliptic BVP
  8. Hölder and quasi-Lipschitz continuity
  9. Gronwall inequality

Key Result

Theorem 3.1

For every $q\in H$, the weak formulation eq:6 has a unique solution $\psi$ in $V$, and the solution satisfies the elliptic PDE in the distribution sense, the homogeneous Neumann boundary conditions at the top and bottom in the sense that and the circulation restrictions along each boundary loop $\Gamma_l$, $0\leq l\leq L$, in the sense that

Figures (1)

  • Figure 1: A multiply-connected horizontal cross-section of the cylindrical domain.

Theorems & Definitions (15)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['l:1']}
  • Lemma 4.3
  • proof
  • proof : Proof of Theorem \ref{['t:1']}
  • Theorem 5.1
  • proof
  • ...and 5 more