Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition

Abstract

This article studies the partial vanishing viscosity limit of the 2D Boussinesq system in a bounded domain with a slip boundary condition. The result is proved globally in time by a logarithmic Sobolev inequality. 2010 MSC: 35Q30; 76D03; 76D05; 76D07.

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.…”

“…Then there exist a unique solution (v, θ) to ( 50)-( 54) and a positive constant Proof. Since it is well known that the solution (v, θ) is smooth see [26]- [24], we need to show estimates (56) and (57). First, by maximum principle, it follows from (51) that…”

“…Using the Gagliardo-Nirenberg inequality, the Hölder inequality, and proceed as [24]- [26], we obtain the following estimate v t (t, •) 2 + 1 2…”

“…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)).…”