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

2012/10/27 by Chongsheng Cao, Slim Ibrahim, Cao, Chongsheng +5 · 6 citations
Engineering · Mathematics · Physics and Astronomy · #35Q35 #65M70 #86-08 #86A10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Atmospheric and Oceanic Physics (physics.ao-ph) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Geophysics (physics.geo-ph) #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations #math.AP #msc:35Q35 #msc:65M70 #msc:86-08 #msc:86A10 #physics.ao-ph #physics.flu-dyn #physics.geo-ph

paper · pdf · doi:10.48550/arxiv.1210.7337

arxiv created 2012/10/27 · openalex publication_date 2012/10/27 · arxiv updated 2012/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

Citations

• Regularity and integrability of 3D Euler and Navier–Stokes equations for rotating fluids

Cited by

• A geophysical free-boundary system modeling an ice-sheet interacting with an ocean
• Global well-posedness of the 2D primitive equations with fractional horizontal dissipation
• On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system
• Rigorous justification of the hydrostatic-incompressible approximation for weakly stratified isothermal flow
• Well-posedness and ill-posedness of the primitive equations with fractional horizontal dissipation
• Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations

Related

• Stable Singularity Formation for the Inviscid Primitive Equations
• Global Well--posedness of the Three-dimensional Viscous Primitive Equations of Large Scale Ocean and Atmosphere Dynamics
• Global well-posedness of the 3D primitive equations with horizontal viscosity and vertical diffusivity II: close to H¹ initial data
• Strong solutions to the 3D primitive equations with only horizontal dissipation: near H¹ initial data
• Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid\n Primitive Equations with Rotation