Global regularity for a two‐dimensional nonlinear Boussinesq system

Abstract

In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi‐strong solutions and get some higher order estimates, and then prove the global well‐posedness of the two‐dimensional Boussinesq system with variable viscosity for H3 initial data. Copyright © 2016 John Wiley & Sons, Ltd.

Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation

“…data, in order to guarantee the assumption (A.1). Our analysis here is close to [30,15,24,35,27,29,26] and we will keep many arguments proved in these papers.…”

“…The boundary condition for the velocity v in (2) is the free boundary condition of the Navier boundary condition. Furthermore, if we add suitable initial data (v 0 , θ 0 ) satisfying divv 0 = 0 and the compatibilty condition (2), then we can prove, see Appendix A and [15,23,24,26,27,29,30,31,35,36], that the initial boundary value problem corresponding to the system (1) possesses a unique solution (v, θ, π) such that v ∈ C([0, T ]; H 3 (Ω)) ∩ L 2 (0, T ; H 4 (Ω)), v t ∈ L 2 (0, T ; L 2 (Ω)), θ ∈ C([0, T ]; H 3 (Ω)), π ∈ L 2 (0, T ; H 1 (Ω)).…”

“…The purpose of this section is to give some regularity properties of the direct problem of the 2D Boussinesq system and we introduce some material and method which can be used to find the assumption (A.2). By taking time derivative to the following system (50)-( 54) and proceeding as the proof of Proposition 2 in below and [29], we can extend the regularity of the solution, provided the regularity of the initial…”

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

“…In the case of anisotropy dissipation, many authors have proved the global well-posedness (see, e.g., [2,3,9,17,42,44,61,80]). For a detailed review on interesting results, we refer the reader to [52,57].…”

A Lagrangian approach for inhomogeneous Boussinesq equations

“…Wang and Zhang obtained the global well‐posedness of the two dimensions Boussinesq system in the case when and depend on the temperature in Wang and Zhang . Qin et al proved the global well‐posedness of two‐dimensional Boussinesq system with variable viscosity when the initial data belongs to in Qin et al, and they also proved the existence of global classical solutions to the three dimensions Boussinesq system in Qin et al…”