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

Abstract

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, if they exist, they 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.MSC Subject Classifications: 35Q35, 65M70, 86-08,86A10.

Global Well‐Posedness of the Three‐Dimensional Primitive Equations with Only Horizontal Viscosity and Diffusion

“…As the counterpart of [6], global well-posedness of strong solutions to the primitive equations with full viscosities but only horizontal diffusion was later obtained in [5], still for H 2 initial data. Notably, smooth solutions to the inviscid primitive equation, with or without coupling to the temperature equation, has been shown [4] to blow up in finite time (see also Wong [27]).…”

Blowup in stagnation-point form solutions of the inviscid 2d Boussinesq equations

“…Moreover, we also show that this blowup implies either one-sided or two-sided blowup in the vorticity. 1 Our blowup criteria is local-inspace and relies both on initial velocities with a local profile characterized by the non-vanishing of f 0 (x) at the boundary and non-negativity of the initial temperature ρ 0 (x). Due to the local nature of the blowup criteria, our results do not rule out the formation of finite-time singularities either in the interior of the domain or at the boundary if f 0 possesses a different local structure.…”

“…(8) is interesting in its own right from a mathematical perspective: it illustrates how the boundary conditions, more particularly periodic or Dirichlet boundary conditions, can either contribute to, or suppress, the formation of spontaneous singularities from smooth initial conditions in nonlinear evolution equations [23]. Moreover, (8) appears as a reduced 1D model for the 3D inviscid primitive equations of large scale oceanic and atmospheric dynamics [1], and is also related to the hydrostatic Euler equations [28,13]. The term 'stagnation-point similitude' arises from the observation that velocity fields of the form (5)i) emerge from the modeling of flow near a stagnation point [26,18,10].…”

Decay rate of unique global solution for a class of 2D tropical climate model

“…For the sake of simplicity, we will limit ourselves to μ = ν = η = 1 in the rest of the paper and thus compliment it with the following initial condition: It should be mentioned that the original equations derived in Frierson et al has no viscous terms in and , at the same time, the Laplacian terms are also not involved, this is because it is derived from the inviscid primitive equations by performing a Galerkin truncation up to the first baroclinic mode. Some mathematical results concerning the primitive equations have been addressed extensively, for which the reader is referred to previous studies …”