Differential and Integral Equations
Volume 16, Number 2, February 2003, Pages 129–158
EQUILIBRIUM SOLUTIONS OF THE BÉNARD
EQUATIONS WITH AN EXTERIOR FORCE
B. Scarpellini
Mathematisches Institut, Universitt Basel
Rheinsprung 21, CH-4051 Basel, Switzerland
(Submitted by: Reza Aftabizadeh)
Abstract. In this paper we investigate questions of existence and
uniqueness of equilibrium solutions of the inhomogeneous Bénard equations with exterior force f which may be generated by a magnetic field
which is not too strong. Two types of results are obtained. The first,
based on a priori estimates and degree arguments states that there is
a certain range λ ≤ λc + δ1 for the Rayleigh parameter λ, for which
existence can be asserted for any given force f , while the second result
says that for λ < λc , λc − λ small, there are forces f which give rise to
three different solutions. Here, λ2c = Rc is the critical Rayleigh number
which enters basically into our considerations.
0. Introduction
Among the much studied equations in fluid dynamics are the Bénard equations (BE) which have given rise to a vast literature (see e.g. [5], [10] for
reviews). The main questions which are usually studied in connection with
the BE are stability questions of various types and questions of global existence in the time dependent case, while existence and nature of bifurcations
are the principal topics in the time independent case. Less attention has
been paid to the time independent BE containing an exterior force as additional term and to the question of the existence of equilibrium solutions. It
is this problem which will be studied in this paper. In fact we consider the
slightly more general system:
ν∆v + λkV + β∂x v + γ∂y v − (v∇)v + f! − ∇p = 0,
κ∆V + λv3 + β∂x V + γ∂y V − (v∇)V + f4 = 0,
div v = 0, k = (0, 0, 1)t , f = (f!, f4 ) = (f1 , f2 , f3 , f4 ).
Accepted for publication: June 2002.
AMS Subject Classifications: 35J60, 35Q30, 35Q35, 76D05.
129
(0.1)