Long‐Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

Jamel Benameur,Mongi Blel

doi:10.1155/2012/627813

Abstract

We study the behavior at infinity in time of any global solution of the surface quasigeostrophic equation with subcritical exponent 2/3 ≤ α ≤ 1. We prove that . Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that if is a global solution, then .

Highlights

We consider the 2D dissipative quasi-geostrophic equation with subcritical exponent 1/2 < α ≤ 1,∂tθ −Δ αθ u · ∇ θ 0 in R × R2, Sα θ 0, x θ0 x in R2, where x ∈ R2, t > 0, θ θ x, t is the unknown potential temperature, and u u1, u2 is the divergence free velocity which is determined by the Riesz transformation of θ in the following way: u1 −R2θ −∂2 −Δ −1/2θ, 1.1u2 R1θ ∂1 −Δ −1/2θ.Abstract and Applied AnalysisThis equation is a two-dimensional model of the 3D incompressible Euler equations, and if α 1, the equation S1 is the 2D Navier-Stokes equation
In 4, Dong and Du study the critical case α 1/2 in the critical space H 1 R2. They prove the global existence if the initial condition is in the critical space H1 R2
In this case 5, Niche and Schonbek prove that if the initial data θ0 is in L2 R2, the L2 norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay

Summary

Introduction

We consider the 2D dissipative quasi-geostrophic equation with subcritical exponent 1/2 < α ≤ 1,. In 4 , Dong and Du study the critical case α 1/2 in the critical space H 1 R2. They prove the global existence if the initial condition is in the critical space H1 R2. In this case 5 , Niche and Schonbek prove that if the initial data θ0 is in L2 R2 , the L2 norm of the solution tends to zero but with no uniform rate, that is, there are solutions with arbitrary slow decay. If θ0 ∈ Lp R2 , with 1 ≤ p ≤ 2, they obtain a uniform decay rate in L2 They consider the solution in other Lq spaces.

Notations and Technical Lemmas

Existence Theorem

Proof of the Main Theorem

Hα dτ a0