Persistence of global well-posedness for the 2D Boussinesq equations with fractional dissipation

Xing Su,Gangwei Wang,Yue Wang

doi:10.1186/s13662-019-2348-1

Abstract

In this paper, we study the IBVP for the 2D Boussinesq equations with fractional dissipation in the subcritical case, and prove the persistence of global well-posedness of strong solutions. Moreover, we also prove the long time decay of the solutions, and investigate the existence of the solutions in Sobolev spaces W^{2,p}({R}^{2})times W^{1,p}({R}^{2}) for some p>2.

Highlights

1 Introduction In this paper, we study the 2D Boussinesq equations with fractional dissipation
Our main focus of the research on the 2D Boussinesq equation has been on the global regularity issue when only fractional dissipation is present
The following is the first main result of this paper, which asserts the global wellposedness of the 2D Boussinesq equations (1)

Summary

Introduction

We study the 2D Boussinesq equations with fractional dissipation. The model reads ut + νΛ2αu + u · ∇u + ∇P = θ e2, div u = 0, θt + κΛ2β θ + u · ∇θ = 0,. Our main focus of the research on the 2D Boussinesq equation has been on the global regularity issue when only fractional dissipation is present. The following is the first main result of this paper, which asserts the global wellposedness of the 2D Boussinesq equations (1). There exists a unique global solution (u(t), θ (t)) of Boussinesq equations (1) such that, for any T > 0, u(t) ∈ C [0, T]; H1+s R2 ∩ L2 [0, T]; H1+s+α R2 ,. Using Lemma 1 and applying fractional embedding theorems together with Young inequality again, we obtain. Lemma 7 Under the assumptions of Theorem 1, the solution of Boussinesq equations (1) is unique. Taking the derivative D = (∂x1 , ∂x2 ) of both sides of (27), and multiplying the result equation array by Dω|Dω|p–2, after integration by parts, we obtain.

Conclusions

Methods