# Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

@article{Cao2012FiniteTimeBF,
title={Finite-Time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics},
author={Chongsheng Cao and S. A. Hoda Ibrahim and Kenji Nakanishi and Edriss S. Titi},
journal={Communications in Mathematical Physics},
year={2012},
volume={337},
pages={473-482}
}
• Published 27 October 2012
• Mathematics
• Communications in Mathematical Physics
In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations, if they exist, they blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three…
Stable Singularity Formation for the Inviscid Primitive Equations
• Mathematics
• 2021
The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev
An Approach to the Primitive Equations for Oceanic and Atmospheric Dynamics by Evolution Equations
• Mathematics
• 2020
The primitive equations for oceanic and atmospheric dynamics are a fundamental model for many geophysical flows. In this chapter we present a summary of an approach to these equations based on the
Global Well-posedness of the 3D Primitive Equations with Only Horizontal Viscosity and Diffusion
• Mathematics
• 2014
In this paper, we consider the initial-boundary value problem of the 3D primitive equations for planetary oceanic and atmospheric dynamics with only horizontal eddy viscosity in the horizontal
Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
• Mathematics
• 2019
The $3D$-primitive equations with only horizontal viscosity are considered on a cylindrical domain $(-h,h)\times G$, $G\subset \mathbb{R}^2$ smooth, with the physical Dirichlet boundary conditions on
Global well-posedness of z-weak solutions to the primitive equations without vertical diffusivity
• Mathematics
Journal of Mathematical Physics
• 2022
In this paper, we consider the initial boundary value problem in a cylindrical domain to the three dimensional primitive equations with full eddy viscosity in the momentum equations but with only

## References

SHOWING 1-10 OF 33 REFERENCES
Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion
• Mathematics
• 2010
The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly
Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics
• Mathematics, Environmental Science
• 2005
In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere
FAST SINGULAR OSCILLATING LIMITS AND GLOBAL REGULARITY FOR THE 3D PRIMITIVE EQUATIONS OF GEOPHYSICS
• Mathematics
• 2000
Fast singular oscillating limits of the three-dimensional "primitive" equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D
Blow-up of unsteady two-dimensional Euler and Navier-Stokes solutions having stagnation-point form
• Mathematics
Journal of Fluid Mechanics
• 1989
The time-dependent form of the classic, two-dimensional stagnation-point solution of the Navier-Stokes equations is considered. If the viscosity is zero, a class of solutions of the initial-value
ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS
• Mathematics
• 1999
The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a
SOME SIMILARITY SOLUTIONS OF THE NAVIER-STOKES EQUATIONS AND RELATED TOPICS
• Mathematics
• 2000
We consider a semilinear equation arising from the Navier- Stokes equations – the governing equations of viscous fluid motion – and related model equations. The solutions of the semilinear equation
On the Hs Theory of Hydrostatic Euler Equations
• Mathematics
• 2012
In this paper we study the two-dimensional hydrostatic Euler equations in a periodic channel. We prove the local existence and uniqueness of Hs solutions under the local Rayleigh condition. This