Abstract
We show well-posedness for the equations describing a new model of slightly compressible fluids. This model was recently rigorously derived in Grandi and Passerini (Geophys Astrophys Fluid Dyn, 2020) from the full set of balance laws and falls in the category of anelastic Navier–Stokes fluids. In particular, we prove existence and uniqueness of global regular solutions in the two-dimensional case for initial data of arbitrary “size”, and for “small” data in three dimensions. We also show global stability of the rest state in the class of weak solutions.
Highlights
The Oberbeck-Boussinesq approximation [1,20] is a very popular model, used to describe convection in a horizontal layer of fluid heated from below [14]
As noticed for instance in [10,11,18,21], thermodynamic variables such as energy and density cannot be a function of T only, but should depend on the pressure p since, otherwise, Gibbs law would be unattended and stability in waves propagation not allowed. For these more general models, one has necessarily to relax the solenoidal condition on v [6,17] to enlarge the region of the non-dimensional parameter space in which formal limits lead to reliable approximations for compressible fluids
In [13], we proposed a new model for thermal convection in a horizontal layer of fluid heated from below with ρ = ρ(T, p)
Summary
The Oberbeck-Boussinesq approximation [1,20] is a very popular model, used to describe convection in a horizontal layer of fluid heated from below [14]. For these more general models, one has necessarily to relax the solenoidal condition on v [6,17] to enlarge the region of the non-dimensional parameter space in which formal limits lead to reliable approximations for compressible fluids Motivated by this important issues, the author jointly with D. The limiting equations derived (see (OB)β) fall, in the isothermal case, in the category of the so called anelastic approximations of the Navier–Stokes equations; see (2.9), where the velocity field is no longer solenoidal but, instead, satisfies ∇ · (ρv) = 0 It is just the latter that makes the new model interesting from the point of view of well-posedness of the corresponding initial-boundary value problem, which constitutes the focus of this article
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