Finite-time blowup for the 2-D viscous primitive equations of ocean and atmosphere dynamic

Abstract

The atmosphere and ocean primitive equation, which are consist of fluid equations, thermodynamic equations and state equation, play a important role for numerical weather precdiction.In an earlier work we have shown the finite-time blowup for the 3-D primitive equations of oceanic and atmospheric dynamics without viscosity by using the method of self-similar solutions.In this paper we show that for certain class of initial data, the corresponding smooth solutions of the 2-D viscous primitive equations of oceanic and atmospheric dynamics blow up in finite time.Specifically, we consider the 2-D viscous primitive equations of oceanic and atmospheric dynamics in a finite area subject to y-direction and periodic boundary conditions subject to x-direction.To overcome the viscous term, we assume the pressure function $ p(x, , y, t) $ be a concave function and reduce the 2-D primitive equations into new equations on the line $ x = 0 $ with initial and boundary conditions; and then prove that the corresponding smooth solutions of the reduced system develop singularities in finite time.