Z. Angew. Math. Phys. (2017) 68:83
2017 Springer International Publishing AG
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Global regularity for a 3D Boussinesq model without thermal diﬀusion
Abstract. In this paper, we consider a modiﬁed three-dimensional incompressible Boussinesq model. The model considered
in this paper has viscosity in the velocity equations, but no diﬀusivity in the temperature equation. To bypass the diﬃculty
caused by the absence of thermal diﬀusion, we make use of the maximal L
regularity for the heat kernel to establish
the global regularity result.
Mathematics Subject Classiﬁcation. 35Q35, 35B65, 76W05.
Keywords. 3D Boussinesq equations, Global regularity, Zero thermal diﬀusion.
The standard 3D incompressible Boussinesq system with zero thermal diﬀusion takes the following form
u +(u ·∇)u − Δu + ∇π = θe
θ +(u ·∇)θ =0,
u(x, 0) = u
(x),θ(x, 0) = θ
where u =(u
(x, t)) ∈ R
is the velocity, π = π(x, t) ∈ R is the scalar pressure,
θ = θ(x, t) ∈ R is the temperature and e
=(0, 0, 1)
. It is well known that Boussinesq system is
widely used to model the dynamics of the ocean or the atmosphere. It arises from the density-dependent
ﬂuid equations by using the so-called Boussinesq approximation which consists in neglecting the density
dependence in all the terms but the one involving the gravity (see e.g. [13,16]).
It is not diﬃcult to establish the local existence and uniqueness of smooth solutions for system (1.5)
with large initial data (see e.g. [4,13]), whether the unique local smooth solution can exist globally is an
outstanding challenging open problem. Based on this diﬃculty, one may resort to study the mechanism of
blowup and structure of possible singularities of smooth solutions to system (1.1). In this direction, many
researchers were devoted to ﬁnding suﬃcient conditions to ensure the smoothness of the solutions; see
[4,6,8,18,22–24] and so forth. For many interesting results on the high-dimensional Boussinesq equations,
we refer the readers to [1,2,9,10,14,21,25].
It should be pointed out that for zero initial temperature θ
,system(1.5) becomes the 3D classical
Navier–Stokes equations. As system (1.5), the regularity of its weak solutions and the existence of global
strong solutions are challenging open problems. Therefore, there is a considerable body of the literature
on the study of the other model equations of the Navier–Stokes equations (see e.g. [3,5,7,8,15,17,20]).
Here we would like to mention that Chae in  proved the global regularity for smooth initial data for
the following modiﬁed Navier–Stokes equations