A limit problem in natural convection

Abstract

Abstract. We discuss a model limit problem which arises as a first step in the mathematical justification of our Boussinesq-type approximation [4], which takes into account dissipative heating in natural convection. We treat a simplified highly non linear system depending on a (perturbation) parameter ε. The main difficulty is that for ε = 0 the velocity is not solenoidal. First we prove that our system has weak solutions for each fixed ε. Moreover, while the chosen perturbation parameter ε tends to zero we show, that we arrive at the usual incompressible case and the standard Boussinesq approximation.2000 Mathematics Subject Classification: 35K55, 35Q35, 76E19, 76E20, 80A1.

The Oberbeck–Boussinesq approximation as a constitutive limit

“…For such fluids the Oberbeck-Boussinesq approximation has been formally derived in [12] with the help of a power series expansion. In [5], [6] a rigorous mathematical justification of a simplified modell has been carried out. However, the mathematical justification starting with the full system from [12] is still lacking.…”

“…From the mathematical point of view the expansion used in [12] is still formal. In [5], [6] a rigorous justification of simplified problems has been given. We refer the reader to [8], [7] for a completely different approach in which singular limits of the full system are discussed.…”