Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

Abstract

Abstract. In this paper we prove the local existence and uniqueness of C 1+γ solutions of the Boussinesq equations with initial data v0, θ0 ∈ C 1+γ , ω0, ∇θ0 ∈ L q for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C 1+γ solutions of the Boussinesq equations. §1. IntroductionThe interactive motion of a passive scalar (e.g. temperature) and atmosphere with an external potential force is modeled by the following Boussinesq equations:In (1) p denotes the scalar pressure of the fluid flow. It is suggested that these equations have strong resemblance with the 3-D Euler equations in many aspects (see e.g. [7]). In particular the problem of finite time blow-up of smooth solutions of the Boussinesq equations is outstanding as in the case of 3-D Euler equations.In [4], authors proved local existence of solutions of the Boussinesq equations in the Sobolev spaces H m (R 2 ), m > 2, and obtained a blow-up criterion of the smooth solutions. The proofs are similar to Kato's [6] and Beale-Kato-Majda's [2] respectively for the 3-D Euler equations.In this paper we extend the previous results to the case of Hölder continuous initial data. We first prove the unique local existence of the solutions of

Global regularity for the 2D Boussinesq equations with partial viscosity terms

“…On the other hand, the regularity/singularity questions of the case of (B) with = = 0 is an outstanding open problem in the mathematical fluid mechanics (see e.g. [3,4,6,11] for studies in this direction). Even the regularity problem for 'partial viscosity cases' (i.e.…”

A posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid–solid heat transfer

“…As discussed in Section 6.2, such regularity results exist only for a limited range of problems for the Boussinesq approximation [7,13,27,14]. Nevertheless, the error representation can be broken down into four terms, each containing one or more of the following errors: discretization, projection, iteration, and transfer.…”

“…Sufficient regularity can be shown for the Boussinesq approximation in R 2 under certain assumptions on the data and the domain [7,13,27,14]. To our knowledge, these results are not known for the coupled system (1) in general domains.…”

Local Well-Posedness and Blowup Criterion of the Boussinesq Equations in Critical Besov Spaces

“…In our previous paper [13], we considered this system and obtained a theorem of local existence in Besov spaces and a blowup criterion in supercritical case. There are several other results of existence and blowup criteria in different kinds of spaces in supercritical cases which have been obtained, see [7], [5], [6] and [21]. In this paper we consider the system for initial data u 0 and ρ 0 in the critical Besov spaces B s0+1 p,1 with s 0 = 2/p.…”