The initial value problem for the Boussinesq equations with data in Lp

Abstract

Existence and uniqueness of solutions in L p'q for the initial value problem for the Boussinesq equations that describe the flow of a viscous incompressible fluid subject to convective heat transfer is demonstrated via extensive use of the imbedding theorem for singular integral operators.

On the Boussinesq Flow with Nondecaying Initial Data

“…See e.g. [6,12,13,14,15,22,23,31,32,33,34] and papers cited there. These results, however, are imposed the integrability condition on the initial data u 0 ; y 0 A L q for some q < y.…”

Global Well-Posedness Issues for the Inviscid Boussinesq System with Yudovich’s Type Data

“…[5,12]). In contrast, in the case when κ = ν = 0, the Boussinesq system exhibits vorticity intensification and the global well-posedness issue remains an unsolved challenging open problem (except if θ 0 is a constant of course) which may be formally compared to the similar problem for the three-dimensional axisymmetric Euler equations with swirl (see e.g.…”

Global Regularity Results of the 2D Boussinesq Equations with Fractional Laplacian Dissipation

“…It is well-known that the system (1.1) with full Laplacian dissipation (namely, α = β = 2) is global wellposed, see, e.g., [7]. In the case of inviscid Boussinesq equations, the global regularity problem turns out to be extremely difficult and remains outstandingly open.…”